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MTL101:: Tutorial 3 :: Linear Algebra (1) (2) (3) (4) (5) (6) (7) (8) Notation: F = R or C, Pn := {f ∈ F[x] : deg f < n} t 1 2 3 1 0 2 5 Find a basis of the row space of the following matrices: 2 3 4, 0 3 1 1 . 3 4 5 3 1 0 1 For i ∈ {1, 2, . . . , m}, define pi : Fm → F by pi (x1 , x2 , . . . , xm ) = xi (the i-th projection). (a) Show that it is a linear transformation. (b) If T : Fm → F is a linear transformation then it is an F-linear combination of the projections, that is, T = a1 p1 + a2 p2 · · · + am pm for a1 , . . . , am ∈ F. (c) Further, show that S : Fm → Fn is a linear transformation if and only if for each i ∈ {1, 2, . . . , n}, the composition S ◦ pi : Fm → F is a linear transformation. (d) If S : Fm → Fn is a linear transformation then S(x1 , x2 , . . . , xm ) = (y1 , y2 , . . . , yn ) where yi = ai1 x1 + ai2 x2 + · · · + aim xm for aij ∈ F with (1 ≤ i, j ≤ m). Find the rank and nullity of the following linear transformations. Also write a basis of the range space in each case. (a) T : F3 → F3 defined by T (x, y, z) = (x + y + z, x − y + z, x + z). (b) Assume that 0 ≤ m ≤ n. T : Fn → Fm defined by T (x1 , x2 , . . . , xn ) = (x1 , x2 , . . . , xm ). Write the matrix representations of the linear operators with respect to the ordered basis B. (a) T : R2 → R2 where T (x, y) = (x, y), B = {(1, 1), (1, −1)}. (b) D : Pn+1 → Pn+1 such that D(a0 + a1 x + · · · + an xn ) = a1 + 2a2 x + · · · + nan xn−1 , B = {1, x, . . . , xn }. x y x+w z 1 0 0 1 0 0 0 0 (c) T : M2 (F) → M2 (F), 7−→ ,B ={ , , , }. z w z+w x 0 0 0 0 1 0 0 1 Suppose dimV = dimW < ∞ and T : V → W is a linear transformation. Show that the following statements are equivalent (a) T is an isomorphism. (b) T is injective (i.e., one to one). (c) kerT = 0. (d) T is surjective (i.e., onto). Suppose m > n. Justify the following statements: (a) There is no one to one (injective) R-linear transformation from Rm to Rn . (b) There is no onto (surjective) R-linear tranformation from Rn to Rm . Find the eigenvalues, eigenvectors and dimension of eigen-spaces of the following operators. (a) T : R2 → R2 with T (x, y) = (x + y, x), (b) T : R2 → R2 with T (x, y) = (y, x), (c) T : R2 → R2 with T (x, y) = (y, −x) (d) T : C2 (C) → C2 (C) with T (x, y) = (y, −x). (e) Cn → Cn defined by (x1 , x2 , . . . , xn ) 7→ (xn , x1 , . . . , xn−1 ). (f) C2 → C2 defined by (z1 , z2 ) 7→ (z1 − 2z2 , z1 + 2z2 ). (To be discussed in the lecture class) Suppose {λ1 , . . . , λt } is the complete set of distinct eigenvalues of a linear operator T : V → V . Denote by Vi the eigenspace of λi . If Bi is a t t P basis of Vi then show that ∪ Bi is a basis of the sum Vi . Conclude that it is a direct sum. i=1 Further, show that T is diagonalizable if and only if i=1 t P Vi = V . Equivalently, show that T is i=1 diagonalizable if and only if dimV = dimV1 + dimV2 + · · · + dimVt . (9) Find a basis B such that [T ]B is a diagonal matrix in case T is diagonalizable. Find P such that [T ]B = P [T ]S P −1 where S is the standard basis in each case. (a) T : C2 → C2 defined by T (x, y) = (y, −x). (b) T : R3 → R3 defined by T (x, y, z) = (5x − 6y − 6z, −x + 4y + 2z, 3x − 6y − 4z). (c) T : C2 → C2 defined by T (x, y) = (x cos θ + y sin θ, −x sin θ + y cos θ). 2 (10) Characteristic polynomial of a matrix is satisfied by the matrix (Cayley-Hamilton). Use it to find (invertibility and) the inverse of the following linear operators. i) (x, y, z) 7→ (x + y + z, x + z, −x + y). ii) (x, y, z) 7→ (x, x + 2y, x + 2y + 3z). (11) Which of the following is an inner product. (a) h(x1 , y1 ), (x2 , y2 )i = x1 x2 + y1 y2 + 3 on R2 over R. (b) h(x1 , y1 ), (x2 , y2 )i = x1 x2 − y1 y2 on R2 over R. (c) h(x1 , y1 ), (x2 , y2 )i = y1 (x1 + 2x2 ) + y2 (2x1 + 5x2 ) on R2 over R. (d) h(x1 , y1 ), (x2 , y2 )i = x1 x2 + y1 y2 on C2 over C. (e) h(x1 , y1 ), (x2 , y2 )i = x1 x̄2 − y1 ȳ2 on C2 over C. (f) If A, B ∈ Mn (C) define hA, Bi = Trace(AB̄). (g) Suppose C[0, 1] is the space of Rcontinuous complex valued functions on the interval [0, 1] 1 and for f, g ∈ C[0, 1], hf, gi := 0 f (t)ḡ(t) dt. a b (12) Suppose A = ∈ M2 (R) is such that a > 0 and det(A) = ad − b2 > 0. Show that b d hX, Y i = X t AY is an inner product on R2 . p (13) Suppose V is an inner product space. Define ||v|| := hv, vi. Show the following statements. (a) (b) (c) (d) (e) ||v|| = 0 if and only if v = 0 For a ∈ F , ||av|| = |a|||v||. ||v + w|| ≤ ||v|| + ||w||. |||v|| − ||w||| ≤ ||v − w||. hv, wi = 0 ⇐⇒ ||v + w||2 = ||v||2 + ||w||2 . (14) Use standard inner product on R2 over R to prove the following statement: “A parallelogram is a rhombus if and only if its diagonals are perpendicular to each other.” (15) Find with respect to the standard inner product of R3 , an orthonormal basis containing (1, 1, 1). (16) Find an orthonormal basis of R 1P3 := {f (x) ∈ R[x] : deg f (x) < 3} with respect to the inner product defined by hf, gi := 0 f (t)g(t)dt. (17) Suppose W is a subspace of the finite dimensional inner product space V . Define W ⊥ := {v ∈ V : hw, vi = 0 ∀ w ∈ W }. Show the following statements. (a) (b) (c) (d) W ⊥ is a subspace of V . W ∩ W⊥ = 0 V = W ⊕ W ⊥. (W ⊥ )⊥ = W . (18) Suppose W = {(x, y) ∈ R2 : x+y = 0}. Find the shortest distance of (a, b) ∈ R2 from W with respect to i) the standard inner product, ii) the inner product defined by h(x1 , y1 ), (x2 , y2 )i = 2x1 x2 + y1 y2 . ::: END:::