2 Linear Transformations - Solutions

1. Addendum: More on Basis and Dimension (This material appears in Section 2 of the Study Guide, but in Chapter 1 of the text.) Theorems about dimension from Chapter 1 • If X ✓ V is a subspace, then dim(X)  dim(V ). Moreover, if dim(V ) < 1, then dim(X) = dim(V ) if and only if X = V . • Let V be a vector space with dim(V ) = n, and let S ✓ V be a set of m distinct vectors in V . – If m < n, then S cannot span V . – If m > n, then S cannot be linearly independent. Challenge: Explain how the following “two-out-of-three theorem” follows from the results above. Two-out-of-three Theorem for Sets: Let V be a vector space, and let S ✓ V be a set of n distinct vectors in V . If any two of the following conditions hold, then all three hold (and S is a basis for V ). (1) S is linearly independent. (2) S spans V . (3) dim(V ) = n. Ex. 1. Let V be a vector space. Is the set {x, 1 + x, 1 2x + x2 } a basis of P2 (R)? Justify your answer. 2. Linear Transformations (References: Comps Study Guide for Linear Algebra Section 2; Damiano & Little, A Course in Linear Algebra, Chapter 2) Linear Transformations: Let V and W be vector spaces and T : V ! W . Write down the condition(s) that T must satisfy in order to be linear. Ex. 2. Let T be a linear transformation from R2 to R2 . Suppose that T (1, 2) = (3, 4) and T (2, 3) = (1, 1). Find T (x, y) for (x, y) 2 R2 . Ex. 3. Let T : V ! W be a linear transformation between vector spaces V and W . Prove that T (~0V ) = ~0W . (Note: This is a very commonly used result, which you should remember.) Kernel (Nullspace): Let T : V ! W be a linear transformation. Write down a definition of the kernel of T . Nullity: The dimension of the kernel of T , dim(ker(T )), is called the nullity of T . Image (Range): Let T : V ! W be a linear transformation. Write down a definition of the image of T . Rank: The dimension of the image of T , dim(Im(T )), is called the rank of T . Matrices: Let A 2 Mm⇥n (R) be an m ⇥ n matrix. Then T : Rn ! Rm defined by T (x) = Ax is linear. In this case, the image of T is the span of the columns of A, and so is called the column space of A. 2 3 1 1 1 1 1 Ex. 4. Let A = 43 3 3 3 35. 5 5 7 7 7 (a) Find a basis for the column space of A. (b) Find a basis for the null space (or kernel) of A. (c) Find the general solution of the equation A~x = ~0. One-to-one: Let T : V ! W . Write down what it means to say that T is one-to-one (injective). Onto: Let T : V ! W . Write down what it means to say that T is onto (surjective). Isomorphism: A linear transformation that is both one-to-one and onto is called an isomorphism. Ex. 5. Let V and W be vector spaces and let T be a linear transformation from V to W . Prove that T is one-to-one if and only if ker(T ) = {~0V }. (Note: This is a commonly used result, which you should remember.) Ex. 6. Let T : V ! V be a linear transformation on a finite dimensional vector space V . Suppose that T is one-to-one. Prove that if {v1 , v2 , . . . vn } is a basis for V then {T (v1 ), T (v2 ), . . . T (vn )} is also a basis for V . Ex. 7. Let V be a vector space and S = {v1 , v2 } be a subset of V . Define T : R2 ! V by T (x1 , x2 ) = 2x1 v1 + 3x2 v2 . (a) Prove that T is linear. (b) Prove that if S is linearly independent then T is one-to-one. (c) Prove that if Span(S) = V then T is onto. One-to-one, Onto, and Dimension: Let T : V ! W be linear. • T is one-to-one if and only if nullity(T ) = 0. • If dim(W ) < 1, T is onto if and only if rank(T ) = dim(W ). Rank-Nullity Theorem (Dimension Theorem): rank(T ) + nullity(T ) = dim(V ) Challenge: Explain how the following “two-out-of-three theorem” follows from the Rank-Nullity Theorem and the two results above. Two-out-of-three Theorem for Linear Transformations: Let T : V ! W be a linear transformation, and suppose that at least one of V, W is finite-dimensional. If any two of the following conditions hold, then all three hold (and so T is an isomorphism). (1) T is one-to-one. (2) T is onto. (3) dim(V ) = dim(W ). Ex. 8. Let A be a 4 ⇥ 6 matrix with real coefficients. (a) Can the columns of A be linearly independent? (b) Prove that the nullity of A is at least 2. (c) Does the equation A~x = ~0 have a unique solution? (d) Assume that the nullity of A is exactly 2. What does this imply about the rank of A? Ex. 9. Suppose that T : Rn ! Rn is a linear map such that kT (x)k = kxk for all x 2 Rn . Prove that T is an isomorphism. Ex. 10. Let P2 = {a + bt + ct : a, b, c 2 R} and T : P2 ! R be defined by T (p) = 2 (a) (b) (c) (d) Prove that T is linear. Find a basis for the kernel (or null space) of T . Find the rank and nullity of T . Is T one-to-one? Is T onto? Justify your answers. 2  p(1) . p0 (1) Additional Problems Ex. 11. Let V be a vector space. Suppose that {u, v, w} is a basis of V . Is the set {u + 2v, v + 2w, u + 2w} also a basis of V ? Justify your answer. Ex. 12. Let V1 and V2 be vector spaces and let T : V1 ! V2 be a linear transformation. For W2 ✓ V2 , let W1 = {v 2 V1 : T (v) 2 W2 }. Show that if W2 is a subspace of V2 then W1 is a subspace of V1 . Ex. 13. Let T : V ! W be a linear transformation between vector spaces V and W . Prove the following. (a) The kernel of T is a subspace of V . (b) The image of T is a subspace of W . 2 3 1 1 1 3 25. Find bases for the column space and null space of A. Ex. 14. Let A = 4 1 1 0 2 2 4 11 Ex. 15. Let T : R3 ! P2 (R) be defined by T (a1 , a2 , a3 ) = (a1 an isomorphism. a2 ) + a2 x + (a1 + a3 )x2 . Prove that T is Ex. 16. Let V and W be vector spaces and let T be a linear transformation from V to W . Suppose that T is one-to-one and {v1 , v2 , . . . vn } is a set of n linearly independent vectors in V . Prove that {T (v1 ), T (v2 ), . . . T (vn )} is linearly independent in W . Ex. 17. Let V and W be vector spaces and T : V ! W be a linear transformation. Suppose that {T (v1 ), T (v2 ), . . . T (vn )} is linearly independent. (a) Prove that {v1 , v2 , . . . vn } is linearly independent. (b) Suppose in addition that Span({v1 , v2 , . . . vn }) = V . Prove that T is one-to-one. Ex. 18. Let V and W be vector spaces and T : V ! W be a linear transformation. For c 2 R with c 6= 0, define a map S : V ! W by S(v) = cT (v). Prove that if T is onto then S is onto as well. Ex. 19. Let V = M2⇥2 (R) be the vector space of 2 ⇥ 2 matrices with real coefficients and W = P2 (R) be the vector space of polynomials of degree less than or equal to 2. Suppose that T : V ! W is a linear transformation. (a) Explain what is meant by the kernel, or null space, of T . (b) What are the possible values of the nullity, or the dimension of the kernel, of T ? Justify your answer. (c) Can T be one-to-one? Can T be onto? Ex. 20. Let T : V ! W be an isomorphism between finite dimensional vector spaces V and W . Let U be a subspace of V and let T (U ) = {T (u) : u 2 U }. Prove that dim(U ) = dim(T (U )).