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(c)2015 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. Math 217: §2.2 Linear Transformations and Geometry Professor Karen Smith Key Definition: A linear transformation T : Rn → Rm is a map (i.e., a function) from Rn to Rm satisfying the following: • T (~x + ~y ) = T (~x) + T (~y ) for all ~x, ~y ∈ Rn (that is, “T respects addition”). • T (a~x) = aT (~x) for all a ∈ R and ~x ∈ Rn (that is, “T respects scalar mutliplication”). 1 4 0 −2 A. Suppose that −→ is a linear transformation. Suppose T ( )= and T ( )= . 0 2 1 −2 1 ? Find it, using only the definition of linear transformation 1. Do we also know the value of T 1 2 2 ? ? T given above. What about T 1 0 R2 T R2 2. Do we know the value of T on any linear combination a~e1 + b~e2 where ei are the standard unit column vectors in R2 ? Find it, using only the definition of linear transformation given above. a )? Prove it. 3. What is T ( b 4. Find the matrix A such that T ~x = A~x. 5. What does your matrix have to do with T (~e1 ) and T (~e2 )? 6. Can you state a general conjecture? Can you prove your conjecture? B. Let ~e1 , . . . , ~en be the standard unit vectors for Rn . 1. If we know the values of a linear transformation T : Rn → Rd on each ~ei , do we know the value for any ~x ∈ Rn ? Why? Discuss with your tablemates. 2. Prove that T (~x) = A~x where A is the d × n matrix formed by the vectors T (~e1 ), . . . T (~en ). This is a crucial idea. Be sure you understand exactly how a linear transformation can be described using matrix multiplication, and how to get the matrix. Linear transformations in geometry C. Let S : R2 → R2 be dilation by a factor of three. 1. Give a geometric reason that S is a linear transformation using the definition. 2. What is the associated matrix A so that S(~v ) = A~v ? 3. What about dilation (or contration) by an arbitrary factor? D. Let L : R2 → R2 be rotation in the counter-clockwise direction by 90◦ (fixing the origin). 1. Give a geometric explanation why L is a linear transformation using the definition. 2. What is the associated matrix A so that L(~v ) = A~v ? 3. What about rotation through an arbitrary angle θ? To write the matrix, you need to remember your high school trig. E. Let M : R2 → R2 be reflection over the x-axis. 1. Show that M is linear by writing down a formula for it explicitly. 2. What about reflection over the line y = x? Is this a linear tranformation? If so, find its matrix. F. Let Q : R2 → R2 be the transformation that stretches vertically by a factor of two and contracts horizontally by a factor of 3. 1. Show that Q is linear by writing down a formula for it explicitly. 2. What about arbitrary (but different) scale factors vertically and horizontally? What happens if they are negative? G. Find the matrix for a vertical shear by 2. What about a horizontal shear by λ? What happens if λ is negative? Use the “standard L” as in the book to analyze this linear transformation. H. Bonus: Think geometrically: Do you think that reflection over an arbitrary line through the origin is a linear transformation? Can you write down its matrix?