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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Introduction to Linear Transformation Math 4A – Xianzhe Dai UCSB April 14 2014 Based on the 2013 Millett and Scharlemann Lectures 1/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere System of Linear Equations: a11 x1 + a12 x2 + . . . + a1n xn = b1 a21 x1 + a22 x2 + . . . + a2n xn = b2 .. . am1 x1 + am2 x2 + . . . + amn xn = bm 2/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere System of Linear Equations: a11 x1 + a12 x2 + . . . + a1n xn = b1 a21 x1 + a22 x2 + . . . + a2n xn = b2 .. . am1 x1 + am2 x2 + . . . + amn xn = bm can be written as matrix equation A~x = ~b. 2/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere System of Linear Equations: a11 x1 + a12 x2 + . . . + a1n xn = b1 a21 x1 + a22 x2 + . . . + a2n xn = b2 .. . am1 x1 + am2 x2 + . . . + amn xn = bm can be written as matrix equation A~x = ~b. New perspective: think of the LHS as a “function/map/transformation”, T (~x ) = A~x . T maps/transforms a vector ~x to another vector A~x . 2/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere System of Linear Equations: a11 x1 + a12 x2 + . . . + a1n xn = b1 a21 x1 + a22 x2 + . . . + a2n xn = b2 .. . am1 x1 + am2 x2 + . . . + amn xn = bm can be written as matrix equation A~x = ~b. New perspective: think of the LHS as a “function/map/transformation”, T (~x ) = A~x . T maps/transforms a vector ~x to another vector A~x . Two very nice properties it enjoys are T (~u + ~v ) = A(~u + ~v ) = A~u + A~v T (c~u ) = A(c~u ) = cA~u = T (~u ) + T (~v ) = cT (~u ) 2/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Linear Transformations Definition A linear transformation is a function T : Rn → Rm with these properties: For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v ) For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ). 3/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Linear Transformations Definition A linear transformation is a function T : Rn → Rm with these properties: For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v ) For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ). Example: Let T : R1 → R1 be defined by T (x) = 5x. 3/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Linear Transformations Definition A linear transformation is a function T : Rn → Rm with these properties: For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v ) For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ). Example: Let T : R1 → R1 be defined by T (x) = 5x. For any u, v ∈ R 1 , T (u + v ) = 5(u + v ) = 5u + 5v = T (u) + T (v ) and for any c ∈ R, T (cu) = 5cu = c5u = cT (u). 3/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Linear Transformations Definition A linear transformation is a function T : Rn → Rm with these properties: For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v ) For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ). Example: Let T : R1 → R1 be defined by T (x) = 5x. For any u, v ∈ R 1 , T (u + v ) = 5(u + v ) = 5u + 5v = T (u) + T (v ) and for any c ∈ R, T (cu) = 5cu = c5u = cT (u). So T is a linear transformation. 3/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Some notes: Most functions are not linear transformations. 4/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Some notes: Most functions are not linear transformations. For example: cos(x + y ) 6= cos(x) + cos(y ). Or (2x)2 6= 2(x 2 ). 4/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Some notes: Most functions are not linear transformations. For example: cos(x + y ) 6= cos(x) + cos(y ). Or (2x)2 6= 2(x 2 ). For any linear transformation T (~0) = ~0 (this rules out function f (x) = x + 5): 4/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Some notes: Most functions are not linear transformations. For example: cos(x + y ) 6= cos(x) + cos(y ). Or (2x)2 6= 2(x 2 ). For any linear transformation T (~0) = ~0 (this rules out function f (x) = x + 5): Take c = 0, then T (~0) = T (0 · ~0) = 0T (~0) = ~0. The two conditions could be written as one: For any vectors ~u , ~v ∈ Rn and real numbers a, b ∈ R, T (a~u + b~v ) = aT (~u ) + bT (~v ) . 4/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Some notes: Most functions are not linear transformations. For example: cos(x + y ) 6= cos(x) + cos(y ). Or (2x)2 6= 2(x 2 ). For any linear transformation T (~0) = ~0 (this rules out function f (x) = x + 5): Take c = 0, then T (~0) = T (0 · ~0) = 0T (~0) = ~0. The two conditions could be written as one: For any vectors ~u , ~v ∈ Rn and real numbers a, b ∈ R, T (a~u + b~v ) = aT (~u ) + bT (~v ) . Example 4/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Important example: 5/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Important example: Let A be any m × n matrix. Define T : Rn → Rm by T (~x ) = A~x . We have already seen that T has what it takes: 5/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Important example: Let A be any m × n matrix. Define T : Rn → Rm by T (~x ) = A~x . We have already seen that T has what it takes: For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = A(~u + ~v ) = A~u + A~v = T (~u ) + T (~v ) 5/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Important example: Let A be any m × n matrix. Define T : Rn → Rm by T (~x ) = A~x . We have already seen that T has what it takes: For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = A(~u + ~v ) = A~u + A~v = T (~u ) + T (~v ) For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = A(c~u ) = c(A~u ) = cT (~u ). 5/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Important example: Let A be any m × n matrix. Define T : Rn → Rm by T (~x ) = A~x . We have already seen that T has what it takes: For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = A(~u + ~v ) = A~u + A~v = T (~u ) + T (~v ) For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = A(c~u ) = c(A~u ) = cT (~u ). A linear transformation defined by a matrix is called a matrix transformation. 5/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Important example: Let A be any m × n matrix. Define T : Rn → Rm by T (~x ) = A~x . We have already seen that T has what it takes: For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = A(~u + ~v ) = A~u + A~v = T (~u ) + T (~v ) For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = A(c~u ) = c(A~u ) = cT (~u ). A linear transformation defined by a matrix is called a matrix transformation. Important Fact Conversely any linear transformation is associated to a matrix transformation (by using bases). 5/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Mona Lisa transformed 6/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Matrix transformations are important and are also cool! 7/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Matrix transformations are important and are also cool! Example 1, a shear: Consider the matrix transformation T : R2 → R2 given by the 2 × 2 matrix 1 32 A= 0 1 7/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Matrix transformations are important and are also cool! Example 1, a shear: Consider the matrix transformation T : R2 → R2 given by the 2 × 2 matrix 1 32 A= 0 1 x1 0 x1 0 For any horizontal vector ~x = T (~x ) = A~x = 1 23 0 1 = x1 + 32 · 0 0 · x1 + 1 · 0 = x1 0 7/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Matrix transformations are important and are also cool! Example 1, a shear: Consider the matrix transformation T : R2 → R2 given by the 2 × 2 matrix 1 32 A= 0 1 x1 0 x1 0 For any horizontal vector ~x = T (~x ) = A~x = 1 23 0 1 = x1 + 32 · 0 0 · x1 + 1 · 0 = x1 0 So T is the identity on horizontal vectors. 7/24 Prelude Linear Transformations Pictorial examples For any vertical vector ~x = 0 x2 Matrix Is Everywhere 8/24 Prelude Linear Transformations Pictorial examples 0 x2 0 x2 For any vertical vector ~x = T (~x ) = A~x = 1 32 0 1 Matrix Is Everywhere = 1 · 0 + 23 · x2 0 · 0 + 1 · x2 = 8/24 Prelude Linear Transformations Pictorial examples 0 x2 0 x2 For any vertical vector ~x = T (~x ) = A~x = 1 32 0 1 Matrix Is Everywhere = 1 · 0 + 23 · x2 0 · 0 + 1 · x2 = 0 x2 + 3 2 x2 0 8/24 Prelude Linear Transformations Pictorial examples 0 x2 0 x2 For any vertical vector ~x = T (~x ) = A~x = 1 32 0 1 Matrix Is Everywhere = 1 · 0 + 23 · x2 0 · 0 + 1 · x2 = 0 x2 + So a vertical vector is pushed perfectly horizontally, a distance times its length: 3 2 x2 0 3 2 8/24 Prelude Linear Transformations Pictorial examples 0 x2 0 x2 For any vertical vector ~x = T (~x ) = A~x = 1 32 0 1 Matrix Is Everywhere = 1 · 0 + 23 · x2 0 · 0 + 1 · x2 = 0 x2 + So a vertical vector is pushed perfectly horizontally, a distance times its length: x2 (0, 1) (3/2, 1) (2, 1) 3 2 x2 0 3 2 (2+3/2, 1) x1 (2, 0) 8/24 Prelude Linear Transformations Pictorial examples Example 2, scaling: Use A= 2 0 0 12 Matrix Is Everywhere 9/24 Prelude Linear Transformations Pictorial examples Example 2, scaling: Use For any vector ~x = T (~x ) = A~x = x1 x2 2 0 0 12 A= x1 x2 2 0 0 12 = Matrix Is Everywhere 2x1 + 0 · x2 0 · x1 + 21 · x2 = 2x1 x2 2 9/24 Prelude Linear Transformations Pictorial examples Example 2, scaling: Use For any vector ~x = T (~x ) = A~x = x1 x2 2 0 0 12 A= x1 x2 2 0 0 12 = Matrix Is Everywhere 2x1 + 0 · x2 0 · x1 + 21 · x2 = 2x1 x2 2 So T stretches horizontally and contracts vertically: 9/24 Prelude Linear Transformations Pictorial examples Example 2, scaling: Use For any vector ~x = T (~x ) = A~x = x1 x2 2 0 0 12 A= x1 x2 2 0 0 12 = Matrix Is Everywhere 2x1 + 0 · x2 0 · x1 + 21 · x2 = 2x1 x2 2 So T stretches horizontally and contracts vertically: x2 (0, 1) x1 (2, 0) 9/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Example 3, reflection through a line: Use 0 1 A= 1 0 10/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Example 3, reflection through a line: Use 0 1 A= 1 0 x1 x1 0 1 x1 0x1 + 1 · x2 x2 T( )=A = = = x2 x2 1 0 x2 1 · x1 + 0 · x2 x1 10/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Example 3, reflection through a line: Use 0 1 A= 1 0 x1 x1 0 1 x1 0x1 + 1 · x2 x2 T( )=A = = = x2 x2 1 0 x2 1 · x1 + 0 · x2 x1 So T exchanges the two coordinates. 10/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Example 3, reflection through a line: Use 0 1 A= 1 0 x1 x1 0 1 x1 0x1 + 1 · x2 x2 T( )=A = = = x2 x2 1 0 x2 1 · x1 + 0 · x2 x1 So T exchanges the two coordinates. Looks like reflection through the line x1 = x2 : 10/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Example 3, reflection through a line: Use 0 1 A= 1 0 x1 x1 0 1 x1 0x1 + 1 · x2 x2 T( )=A = = = x2 x2 1 0 x2 1 · x1 + 0 · x2 x1 So T exchanges the two coordinates. Looks like reflection through the line x1 = x2 : x2 (s, s) reflect tcelfer x1 10/24 Prelude Linear Transformations Example 4, rotation: Use Pictorial examples A= cos θ − sin θ sin θ cos θ Matrix Is Everywhere 11/24 Prelude Linear Transformations Example 4, rotation: Use Pictorial examples A= cos θ − sin θ sin θ cos θ Matrix Is Everywhere 1 1 cos θ − sin θ 1 cos θ T( )=A = = 0 0 sin θ cos θ 0 sin θ 11/24 Prelude Linear Transformations Example 4, rotation: Use Pictorial examples A= cos θ − sin θ sin θ cos θ Matrix Is Everywhere 1 1 cos θ − sin θ 1 cos θ T( )=A = = 0 0 sin θ cos θ 0 sin θ So horizontal unit vector is rotated c-clockwise an angle θ. 11/24 Prelude Linear Transformations Example 4, rotation: Use Pictorial examples A= cos θ − sin θ sin θ cos θ Matrix Is Everywhere 1 1 cos θ T( )=A = 0 0 sin θ So horizontal unit vector is rotated − sin θ 1 cos θ = cos θ 0 sin θ c-clockwise an angle θ. 0 Similarly, for the vertical unit vector , 1 11/24 Prelude Linear Transformations Pictorial examples Example 4, rotation: Use A= cos θ − sin θ sin θ cos θ Matrix Is Everywhere 1 1 cos θ T( )=A = 0 0 sin θ So horizontal unit vector is rotated − sin θ 1 cos θ = cos θ 0 sin θ c-clockwise an angle θ. 0 Similarly, for the vertical unit vector , so all of plane rotates: 1 R (v) v 11/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Some types of problems that can come up: 12/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Some types of problems that can come up: Question: Suppose T : Rn → Rm is a matrix linear transformation. Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u )? 12/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Some types of problems that can come up: Question: Suppose T : Rn → Rm is a matrix linear transformation. Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u )? Answer: Just do matrix-vector multiplication A~u . The result is a vector in Rm . 12/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Some types of problems that can come up: Question: Suppose T : Rn → Rm is a matrix linear transformation. Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u )? Answer: Just do matrix-vector multiplication A~u . The result is a vector in Rm . Question: Suppose T : Rn → Rm is a matrix linear transformation. Suppose A is the matrix of T and ~b ∈ Rm is given. 12/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Some types of problems that can come up: Question: Suppose T : Rn → Rm is a matrix linear transformation. Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u )? Answer: Just do matrix-vector multiplication A~u . The result is a vector in Rm . Question: Suppose T : Rn → Rm is a matrix linear transformation. Suppose A is the matrix of T and ~b ∈ Rm is given. Find a vector ~u so that T (~u ) = ~b. 12/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Some types of problems that can come up: Question: Suppose T : Rn → Rm is a matrix linear transformation. Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u )? Answer: Just do matrix-vector multiplication A~u . The result is a vector in Rm . Question: Suppose T : Rn → Rm is a matrix linear transformation. Suppose A is the matrix of T and ~b ∈ Rm is given. Find a vector ~u so that T (~u ) = ~b. Answer: Solve the system of equations given by A~x = ~b. Any solution is such a vector ~u . 12/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Some types of problems that can come up: Question: Suppose T : Rn → Rm is a matrix linear transformation. Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u )? Answer: Just do matrix-vector multiplication A~u . The result is a vector in Rm . Question: Suppose T : Rn → Rm is a matrix linear transformation. Suppose A is the matrix of T and ~b ∈ Rm is given. Find a vector ~u so that T (~u ) = ~b. Answer: Solve the system of equations given by A~x = ~b. Any solution is such a vector ~u . Reminder: There may be no solution or exactly one solution or a parameterized family of solutions 12/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere iClicker question Suppose T is a matrix linear transformation with matrix A below, and we are seeking all vectors ~u so that T (~u ) = ~b. 1 2 3 6 ~b = 7 A = 0 4 5 0 0 0 8 How many solutions are there? A) B) C) D) Zero. One Infinity Can’t tell 13/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere iClicker question Suppose T is a matrix linear transformation with matrix A below, and we are seeking all vectors ~u so that T (~u ) = ~b. 1 2 3 6 ~b = 7 A = 0 4 5 0 0 0 8 How many solutions are there? A) B) C) D) Zero. One Infinity Can’t tell 6 What if ~b = 7? 0 13/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Question: Describe all vectors ~u so that T (~u ) = ~b. 14/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Question: Describe all vectors ~u so that T (~u ) = ~b. Answer: This is the same as finding all vectors ~u so that A~u = ~b. Could be no ~u , could be exactly one ~u , or could be a parametrized family of such ~u ’s. 14/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Question: Describe all vectors ~u so that T (~u ) = ~b. Answer: This is the same as finding all vectors ~u so that A~u = ~b. Could be no ~u , could be exactly one ~u , or could be a parametrized family of such ~u ’s. Recall the idea: row reduce the augmented matrix [A : ~b] to merely echelon form. 14/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Question: Describe all vectors ~u so that T (~u ) = ~b. Answer: This is the same as finding all vectors ~u so that A~u = ~b. Could be no ~u , could be exactly one ~u , or could be a parametrized family of such ~u ’s. Recall the idea: row reduce the augmented matrix [A : ~b] to merely echelon form. Augmentation column is pivot column ⇐⇒ no solutions. Augmentation column is only non-pivot column ⇐⇒ unique solution. There are free variables ⇐⇒ there is a parameterized family of solutions whose dimension is the number of free variables. 14/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Question: Describe all vectors ~u so that T (~u ) = ~b. Answer: This is the same as finding all vectors ~u so that A~u = ~b. Could be no ~u , could be exactly one ~u , or could be a parametrized family of such ~u ’s. Recall the idea: row reduce the augmented matrix [A : ~b] to merely echelon form. Augmentation column is pivot column ⇐⇒ no solutions. Augmentation column is only non-pivot column ⇐⇒ unique solution. There are free variables ⇐⇒ there is a parameterized family of solutions whose dimension is the number of free variables. If there are solutions, reduced echelon form makes it easy to describe them. 14/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Definition A linear transformation is a function T : Rn → Rm with these properties: For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v ) For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ). 15/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Definition A linear transformation is a function T : Rn → Rm with these properties: For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v ) For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ). For any linear transformation T (~0) = ~0 15/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Definition A linear transformation is a function T : Rn → Rm with these properties: For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v ) For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ). For any linear transformation T (~0) = ~0 T (a~u + b~v ) = aT (~u ) + bT (~v ) 15/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Definition A linear transformation is a function T : Rn → Rm with these properties: For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v ) For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ). For any linear transformation T (~0) = ~0 T (a~u + b~v ) = aT (~u ) + bT (~v ) This has important implications: if you know T (~u ) and T (~v ) , then you know the values of T on all the linear combinations of ~u and ~v . 15/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Definition A linear transformation is a function T : Rn → Rm with these properties: For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v ) For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ). For any linear transformation T (~0) = ~0 T (a~u + b~v ) = aT (~u ) + bT (~v ) This has important implications: if you know T (~u ) and T (~v ) , then you know the values of T on all the linear combinations of ~u and ~v . Matrix transformation: Let A be any m × n matrix. Define T : Rn → Rm by T (~x ) = A~x . 15/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Matrix Is Everywhere Example: Suppose T : R2 → R2 is a linear transformation so that 1 5 0 3 T( )= ; T( )= 0 2 1 4 16/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Matrix Is Everywhere Example: Suppose T : R2 → R2 is a linear transformation so that 1 5 0 3 T( )= ; T( )= 0 2 1 4 −1 What is T ( )? 7 16/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Matrix Is Everywhere Example: Suppose T : R2 → R2 is a linear transformation so that 1 5 0 3 T( )= ; T( )= 0 2 1 4 −1 What is T ( )? 7 −1 1 0 1 0 T( ) = T (−1 +7 ) = −1T ( ) + 7T ( ) 7 0 1 0 1 5 3 16 = −1 +7 = 2 4 26 16/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Matrix Is Everywhere Example: Suppose T : R2 → R2 is a linear transformation so that 1 5 0 3 T( )= ; T( )= 0 2 1 4 −1 What is T ( )? 7 −1 1 0 1 0 T( ) = T (−1 +7 ) = −1T ( ) + 7T ( ) 7 0 1 0 1 5 3 16 = −1 +7 = 2 4 26 In fact, nothing can stop us from using the same idea to compute 2 T( ) or T (~x ) for any vector ~x ∈ R2 : −4 Example 16/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere We can carry this much further: All linear transformations T : Rn → Rm are matrix linear transformations 17/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere We can carry this much further: All linear transformations T : Rn → Rm are matrix linear transformations Why? 0 0 1 x1 0 1 0 x2 ~x = . = x1 . + x2 . + . . . xn . . . . .. . . . 1 0 0 xn is a linear combination of the vectors 1 0 0 0 1 0 .. , .. , . . . , .. . . . 0 0 1 (standard basis of Rn ) 17/24 Prelude Linear Transformations Pictorial examples So by the property of linear transformation 0 1 1 0 T (~x ) = x1 T . + x2 T . + . . . xn T . .. . 0 0 Matrix Is Everywhere 0 0 .. . 1 only need to know each T (~ej ) where 0 .. . th ~ej = 1 ← j entry .. . 0 18/24 Prelude Linear Transformations Pictorial examples So by the property of linear transformation 0 1 1 0 T (~x ) = x1 T . + x2 T . + . . . xn T . .. . 0 0 Matrix Is Everywhere 0 0 .. . 1 only need to know each T (~ej ) where 0 .. . th ~ej = 1 ← j entry .. . 0 Denote a~j = T (~ej ) T (~x ) = x1 a~1 + x2 a~2 + . . . xn a~n 18/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Ahha: In matrix notation this is written: T (~x ) = ~a1 ~a2 x1 x2 . . . ~an . = A~x . . xn 19/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Ahha: In matrix notation this is written: T (~x ) = ~a1 ~a2 x1 x2 . . . ~an . = A~x . . xn That is, the matrix A = ~a1 ~a2 . . . ~an is the matrix of T ! 19/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Recap: 20/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Recap: Since T is a linear transformation, T (~x ) = T (x1~e1 +x2~e2 +. . . xn~en ) = x1 T (~e1 )+x2 T (~e2 )+. . . xn T (~en ) = 20/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Recap: Since T is a linear transformation, T (~x ) = T (x1~e1 +x2~e2 +. . . xn~en ) = x1 T (~e1 )+x2 T (~e2 )+. . . xn T (~en ) = x1~a1 + x2~a2 + . . . xn~an = ~a1 ~a2 . . . ~an x1 x2 .. . = A~x . xn 20/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Recap: Since T is a linear transformation, T (~x ) = T (x1~e1 +x2~e2 +. . . xn~en ) = x1 T (~e1 )+x2 T (~e2 )+. . . xn T (~en ) = x1~a1 + x2~a2 + . . . xn~an = ~a1 ~a2 . . . ~an x1 x2 .. . = A~x . xn Example 20/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rm is a (matrix) linear transformation. Definition T is 1 to 1 if ~u 6= ~v implies that T (~u ) 6= T (~v ). 21/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rm is a (matrix) linear transformation. Definition T is 1 to 1 if ~u 6= ~v implies that T (~u ) 6= T (~v ). Put another words, T (~u ) = T (~v ) implies ~u = ~v . 21/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rm is a (matrix) linear transformation. Definition T is 1 to 1 if ~u 6= ~v implies that T (~u ) 6= T (~v ). Put another words, T (~u ) = T (~v ) implies ~u = ~v . If T (~u ) = T (~v ) we see that T (~u − ~v ) = ~0 (T linear transformation). 21/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rm is a (matrix) linear transformation. Definition T is 1 to 1 if ~u 6= ~v implies that T (~u ) 6= T (~v ). Put another words, T (~u ) = T (~v ) implies ~u = ~v . If T (~u ) = T (~v ) we see that T (~u − ~v ) = ~0 (T linear transformation). Saying that T is 1 to 1 is the same as saying that ~ = ~0 (only trivial solution). T (~ w ) = ~0 exactly when w 21/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rm is a (matrix) linear transformation. Definition T is 1 to 1 if ~u 6= ~v implies that T (~u ) 6= T (~v ). Put another words, T (~u ) = T (~v ) implies ~u = ~v . If T (~u ) = T (~v ) we see that T (~u − ~v ) = ~0 (T linear transformation). Saying that T is 1 to 1 is the same as saying that ~ = ~0 (only trivial solution). T (~ w ) = ~0 exactly when w This means that the reduced echelon form of the matrix of T must have exactly n non-zero rows. 21/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rm is a (matrix) linear transformation. Definition T is 1 to 1 if ~u 6= ~v implies that T (~u ) 6= T (~v ). Put another words, T (~u ) = T (~v ) implies ~u = ~v . If T (~u ) = T (~v ) we see that T (~u − ~v ) = ~0 (T linear transformation). Saying that T is 1 to 1 is the same as saying that ~ = ~0 (only trivial solution). T (~ w ) = ~0 exactly when w This means that the reduced echelon form of the matrix of T must have exactly n non-zero rows. The only solution to the homogeneous equation is the zero solution. And, as a consequence, n ≤ m. 21/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rm is a (matrix) linear transformation. Definition T is 1 to 1 if ~u 6= ~v implies that T (~u ) 6= T (~v ). Put another words, T (~u ) = T (~v ) implies ~u = ~v . If T (~u ) = T (~v ) we see that T (~u − ~v ) = ~0 (T linear transformation). Saying that T is 1 to 1 is the same as saying that ~ = ~0 (only trivial solution). T (~ w ) = ~0 exactly when w This means that the reduced echelon form of the matrix of T must have exactly n non-zero rows. The only solution to the homogeneous equation is the zero solution. And, as a consequence, n ≤ m. Examples: shears, contractions and expansions, rotations, reflections 21/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rm is a (matrix) linear transformation. Definition T is onto if, for any ~v ∈ Rm , is there a ~u ∈ Rn such thatT (~u ) = ~v . 22/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rm is a (matrix) linear transformation. Definition T is onto if, for any ~v ∈ Rm , is there a ~u ∈ Rn such thatT (~u ) = ~v . Saying that T is onto is the same as saying that T (~u ) = ~v always has a solution. 22/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rm is a (matrix) linear transformation. Definition T is onto if, for any ~v ∈ Rm , is there a ~u ∈ Rn such thatT (~u ) = ~v . Saying that T is onto is the same as saying that T (~u ) = ~v always has a solution. This means that the reduced echelon form of the matrix of T must have exactly m non-zero rows. 22/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rm is a (matrix) linear transformation. Definition T is onto if, for any ~v ∈ Rm , is there a ~u ∈ Rn such thatT (~u ) = ~v . Saying that T is onto is the same as saying that T (~u ) = ~v always has a solution. This means that the reduced echelon form of the matrix of T must have exactly m non-zero rows. The non-homogeneous equation must always have a solution. And, as a consequence, m ≤ n. 22/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rm is a (matrix) linear transformation. Definition T is onto if, for any ~v ∈ Rm , is there a ~u ∈ Rn such thatT (~u ) = ~v . Saying that T is onto is the same as saying that T (~u ) = ~v always has a solution. This means that the reduced echelon form of the matrix of T must have exactly m non-zero rows. The non-homogeneous equation must always have a solution. And, as a consequence, m ≤ n. Examples: shears, contractions and expansions, rotations, reflections 22/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rn is a (matrix) linear transformation. Definition T is an isomorphism if T is both 1 to 1 and onto. 23/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rn is a (matrix) linear transformation. Definition T is an isomorphism if T is both 1 to 1 and onto. Saying that T is isomorphism is the same as saying that T is a bijection that respects the vector space structure. 23/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rn is a (matrix) linear transformation. Definition T is an isomorphism if T is both 1 to 1 and onto. Saying that T is isomorphism is the same as saying that T is a bijection that respects the vector space structure. This means that the reduced echelon form of the matrix of T must have exactly n non-zero rows, the same as the number of columns. 23/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rn is a (matrix) linear transformation. Definition T is an isomorphism if T is both 1 to 1 and onto. Saying that T is isomorphism is the same as saying that T is a bijection that respects the vector space structure. This means that the reduced echelon form of the matrix of T must have exactly n non-zero rows, the same as the number of columns. The non-homogeneous equation must always have exactly one solution. 23/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Suppose T : Rn → Rn is a (matrix) linear transformation. Definition T is an isomorphism if T is both 1 to 1 and onto. Saying that T is isomorphism is the same as saying that T is a bijection that respects the vector space structure. This means that the reduced echelon form of the matrix of T must have exactly n non-zero rows, the same as the number of columns. The non-homogeneous equation must always have exactly one solution. Examples: shears, contractions and expansions, rotations, reflections 23/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Summarizing Suppose T : Rn → Rm is a (matrix) linear transformation. 24/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Summarizing Suppose T : Rn → Rm is a (matrix) linear transformation. T is 1 to 1 if there is a pivot 1 in every column of the reduced echelon form, i.e. there are no free variables. 24/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Summarizing Suppose T : Rn → Rm is a (matrix) linear transformation. T is 1 to 1 if there is a pivot 1 in every column of the reduced echelon form, i.e. there are no free variables. Said differently, the column vectors of the matrix of T are linearly independent. 24/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Summarizing Suppose T : Rn → Rm is a (matrix) linear transformation. T is 1 to 1 if there is a pivot 1 in every column of the reduced echelon form, i.e. there are no free variables. Said differently, the column vectors of the matrix of T are linearly independent. T is onto if there is a pivot 1 in every row of the reduced echelon form. 24/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Summarizing Suppose T : Rn → Rm is a (matrix) linear transformation. T is 1 to 1 if there is a pivot 1 in every column of the reduced echelon form, i.e. there are no free variables. Said differently, the column vectors of the matrix of T are linearly independent. T is onto if there is a pivot 1 in every row of the reduced echelon form. Said differently, the column vectors of the matrix of T span the whole space Rm . 24/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Summarizing Suppose T : Rn → Rm is a (matrix) linear transformation. T is 1 to 1 if there is a pivot 1 in every column of the reduced echelon form, i.e. there are no free variables. Said differently, the column vectors of the matrix of T are linearly independent. T is onto if there is a pivot 1 in every row of the reduced echelon form. Said differently, the column vectors of the matrix of T span the whole space Rm . T is an isomorphism if there is a pivot 1 in every row and column, i.e. the reduced echelon matrix is the identity matrix. 24/24 Prelude Linear Transformations Pictorial examples Matrix Is Everywhere Summarizing Suppose T : Rn → Rm is a (matrix) linear transformation. T is 1 to 1 if there is a pivot 1 in every column of the reduced echelon form, i.e. there are no free variables. Said differently, the column vectors of the matrix of T are linearly independent. T is onto if there is a pivot 1 in every row of the reduced echelon form. Said differently, the column vectors of the matrix of T span the whole space Rm . T is an isomorphism if there is a pivot 1 in every row and column, i.e. the reduced echelon matrix is the identity matrix. Said differently, the column vectors of the matrix of T are linearly independent and span the whole space. 24/24