Tutorial Sheet 3 PDF
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This tutorial sheet covers various aspects of linear transformations, including finding linear transformations, examining their properties, and calculating their matrix representations. It involves problems about finding linear transformations (LTs) and their inverses, determining whether given transformations are linear or not, and finding the range and null space of linear transformations.
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(Linear Transformation) 1(i). Find a LT T : R2 → R2 such that T (1, 0) = (1, 1) and T (1, 1) = (−1, 2). Also prove that T maps square with vertices at (0, 0), (1, 0), (1, 1), (0, 1) into a parallelogram. 1(ii). If possible, find a LT T : A → B such that (a) T (2, 3) = (4, 5), T (1, 0) = (0, 0), wh...
(Linear Transformation) 1(i). Find a LT T : R2 → R2 such that T (1, 0) = (1, 1) and T (1, 1) = (−1, 2). Also prove that T maps square with vertices at (0, 0), (1, 0), (1, 1), (0, 1) into a parallelogram. 1(ii). If possible, find a LT T : A → B such that (a) T (2, 3) = (4, 5), T (1, 0) = (0, 0), where A = R2 and B = R2. (b) T (1, 1) = (1, 0, 1), T (0, 1) = (1, 0, 0), T (1, 2) = (2, 1, 1) where A = R2 and B = R3. (c) T (1, 0, 0) = (2, 3), T (0, 1, 0) = (1, 2), T (0, 0, 1) = (−1, −4) where A = R3 and B = R2. (d) T (1, 1, 0) = (0, 1, 1), T (0, 0, 0) = (0, 0, 1), T (1, 0, 1) = (0, 0, 0) where A = B = R3. 2(i). Find a LT T : R3 → R3 , whose range is spanned by the vectors (1, 0, −1) and (1, 2, 2). 2(ii). Find a nonzero LT T : R2 → R2 , which maps all the vectors on the line y = x onto the origin. 3. Find the range and null space of followings LTs. Also find the rank and nullity wherever applicable: (i) T : R2 → R3 defined by T (x1 , x2 ) = (3x1 + x2 , 0, 0). (ii) T : R4 → R3 defined by T (x1 , x2 , x3 , x4 ) = (x1 − x4 , x2 + x3 , x3 − x4 ). (iii) T : R2 → R2 defined by T (x1 , x2 ) = (x1 + x2 , x1 + x2 ). (iv) T : P3 → R3 defined by T (a0 + a1 x + a2 x2 + a3 x3 ) = (a0 + a1 + 2a3 , 2a1 + a2 , a3 + a1 ). (v) T : C(0, 1) → C(0, 1) defined by T (f )x = f (x) sin x. 4. Examine whether the following transformations are linear or not. In case of LT, find their matrix representation with respect to given bases B1 and B2. (i) T : R2 → R2 defined by T (x1 , x2 ) = (x1 + x2 , x2 ); B1 and B2 are standard bases. (ii) T : R2 → R3 defined by T (x1 , x2 ) = (x1 , x1 + x2 , x2 ); B1 and B2 are standard bases. (iii) T : C2 → C2 defined by T (x1 + ix2 , x3 + ix4 ) = (x1 , x2 ); B1 = {(0, 1), (1, 1)} and B2 is standard bases. (iv) T : P2 → P2 defined by T (a0 + a1 x + a2 x2 ) = −a0 + 2a1 x + (a2 + a0 )x2 ; B1 and B2 are standard bases. (v) T : P3 → P3 defined by T (a0 + a1 x + a2 x2 + a3 x3 ) = a0 + a1 (x + 1) + a2 (x + 1)2 + a3 (x + 1)3 ; B2 = {1, 1 + x, 1 + x2 , 1 + x3 } and B1 is standard basis. Rx (vi) T : P2 → P3 defined by T (p(x)) = xp(x) + 0 p(t); B1 and B2 are standard bases. (vii) T : P2 → R4 defined by T (a0 + a1 x + a2 x2 ) = (a0 + a2 , a1 − a0 , a2 − a1 , a0 ); B1 = {1; 1 + x; x + x2 } and B2 = {(1, 0, 1, 0); (1, 0, 0, 0); (0, 1, −1, 0); (0, 0, 1, 1)}. 2×2 2×2 2×2 1 1 (viii) T : R →R defined by T (A) = AM, ∀A ∈ R , where M = is a fixed matrix in R2×2 ; B1 and 1 2 B2 are standard bases. (ix) Repeat part (viii), when T : R2×2 → R2×2 is defined by T (A) = A + M. 5. Let T : R3 → R3 be defined by T (x1 , x2 , x3 ) = (x1 + x2 , x1 + 2x2 , 3x3 + x2 ). Show that T is invertible and further, find a formula for T −1. Match the result by matrix representation also. 7 2 0 0 6(i). Find a LT T : R3 → R3 , whose matrix representation is 2 −5 0 , with respect to standard bases. Find 0 2 1 its inverse matrix also. 1 0 −1 6(ii). Find a LT T : R3 → R3 , whose matrix representation is 0 2 1 , with respect to standard bases. Find 1 0 −1 the matrix of T with respect to basis {(1, 1, −1), (1, 2, 0), (1, 0, 1)}. 1 2 3 0 6(iii). Find a LT T : P3 → R3 , whose matrix representation is 0 1 1 1 , with respect to {1; 1 + x2 ; x + 5 4 1 −1 x3 ; 1 + x + x2 } and {(1, 0, 1), (2, 4, 5), (0, 0, 1)}. 8
