Solution: (Calculus required) For Exercises 6368, some basic knowledge ofcalculus is

Let T : V R2 be a linear transformation satisfying T(v1) = 1 2  , T(v2) = 3 1  Find T(v2 2v1).

Let T : V P2 be a linear transformation satisfying T(v1) = x2 + 1 T(v2) = x2 + 3x T(v3) = 4x 2 Find T(2v1 + v2 3v3).

Let T : P2 R2 be a linear transformation satisfying T(x2 + 1) = 1 3  , T(4x + 3) = 2 1  Find T(2x2 4x 1). (HINT: Write 2x2 4x 1 as a linear combination of x2 + 1 and 4x + 3.)

Let T : R22 P2 be a linear transformation satisfying T 1 2 1 3 = x2 x + 3 T 2 4 1 1  = 3x2 + 4x 1 Find T 7 2 1 7. (HINT: Write 7 2 1 7 as a linear combination of 1 2 1 3 and 2 4 1 1  .)

For Exercises 510, prove that the given function is a linear transformation. T : P2 P2 with T(ax2 + bx + c) = c x2 + bx + a

For Exercises 510, prove that the given function is a linear transformation. T : C[0, 1] R2 with T( f ) = f (0) f (1)

For Exercises 510, prove that the given function is a linear transformation. T : Pn C[0, 1] with T(p(x)) = e x p(x)

For Exercises 510, prove that the given function is a linear transformation. T : R22 P2 with T a b c d = (a d)x2 bx + c

For Exercises 510, prove that the given function is a linear transformation. T : R22 R with T(A) = tr(A) (the trace of A).

For Exercises 510, prove that the given function is a linear transformation. T : V W with T(v) = 0W for all v.

For Exercises 1122, determine if the given function is a linear transformation. Be sure to completely justify your answer. T : Rn Rn with T(v) = 4v

For Exercises 1122, determine if the given function is a linear transformation. Be sure to completely justify your answer. : R2 R2 with T a b  = a b  + 3 2 

For Exercises 1122, determine if the given function is a linear transformation. Be sure to completely justify your answer. T : R2 R2 with T a b  = b a 

For Exercises 1122, determine if the given function is a linear transformation. Be sure to completely justify your answer. T : Rn Rn with T(v) = 0 . . . 0

For Exercises 1122, determine if the given function is a linear transformation. Be sure to completely justify your answer. T : P2 R2 with T(ax2 + bx + c) = a b b + c

For Exercises 1122, determine if the given function is a linear transformation. Be sure to completely justify your answer. T : R22 R with T(A) = det(A)

For Exercises 1122, determine if the given function is a linear transformation. Be sure to completely justify your answer. T: Rnn Rnn with T(A) = AT

For Exercises 1122, determine if the given function is a linear transformation. Be sure to completely justify your answer. T : C[0, 1] R with T( f ) = f (2)

For Exercises 1122, determine if the given function is a linear transformation. Be sure to completely justify your answer. T : Rnn R with T(A) = tr(A)

For Exercises 1122, determine if the given function is a linear transformation. Be sure to completely justify your answer. T : C[0, 1] C[0, 1] with T( f ) = x + f (x)

For Exercises 1122, determine if the given function is a linear transformation. Be sure to completely justify your answer. T : R32 R22 with T(A) = AT A

For Exercises 1122, determine if the given function is a linear transformation. Be sure to completely justify your answer. T : R2 C[0, 1] with T a b  = aebx I

In Exercises 2326, describe the kernel and range of the given linear transformation. T : P1 R with T(ax + b) = a b

In Exercises 2326, describe the kernel and range of the given linear transformation. T : P1 P2 with T( f ) = x f (x)

In Exercises 2326, describe the kernel and range of the given linear transformation. T : P2 R22 with T(ax2 + bx + c) = a b b c

In Exercises 2326, describe the kernel and range of the given linear transformation. T : C[0, 1] P2 with T( f ) = f (0)x2 + f (1)

In Exercises 2730, determine if the given linear transformation is onto and/or one-to-one. T : P2 R2 with T( f ) = f (1) f (2)

In Exercises 2730, determine if the given linear transformation is onto and/or one-to-one. T : P2 P3 with T( f ) = x f (x)

In Exercises 2730, determine if the given linear transformation is onto and/or one-to-one. T : C[0, 1] R with T( f ) = f (1)

In Exercises 2730, determine if the given linear transformation is onto and/or one-to-one. T : R2 P2 with T a b  = ax2 + (b a)x + (a b) FI

FIND AN EXAMPLE For Exercises 3138, find an example of vector spaces V and W and a function T : V W that meets the given specifications. T is a linear transformation that is one-to-one but not onto.

FIND AN EXAMPLE For Exercises 3138, find an example of vector spaces V and W and a function T : V W that meets the given specifications. T is a linear transformation that is onto but not one-to-one.

FIND AN EXAMPLE For Exercises 3138, find an example of vector spaces V and W and a function T : V W that meets the given specifications. . T is a linear transformation that is neither onto nor one-toone.

FIND AN EXAMPLE For Exercises 3138, find an example of vector spaces V and W and a function T : V W that meets the given specifications. T is a linear transformation that is both onto and one-to-one.

FIND AN EXAMPLE For Exercises 3138, find an example of vector spaces V and W and a function T : V W that meets the given specifications. T is a linear transformation such that dim ker(T)  = 1 and dim range(T)  = 3.

FIND AN EXAMPLE For Exercises 3138, find an example of vector spaces V and W and a function T : V W that meets the given specifications. T is a linear transformation such that dim ker(T)  = 4 and dim range(T)  = 2.

FIND AN EXAMPLE For Exercises 3138, find an example of vector spaces V and W and a function T : V W that meets the given specifications. T is a linear transformation such that, for any set of linearly independent vectors {v1, ... , vk } the set {T(v1), ... , T(vk )} is linearly dependent.

FIND AN EXAMPLE For Exercises 3138, find an example of vector spaces V and W and a function T : V W that meets the given specifications. T satisfies condition (b) but not condition (a) of Definition 9.1.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. If T : V W is a linear transformation, then T(v1 v2) = T(v1) T(v2).

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. If T : V W is a linear transformation, then T(v) = 0W implies that v = 0V .

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. . If T : V W is a linear transformation and S is a subspace of V, then T(S) is a subspace of W.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. . If T : V W is a linear transformation and S is a subspace of V, then T(S) is a subspace of W.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. If T : V W is a linear transformation and {v1, ... , vk } is a linearly independent set, then so is {T(v1), ... , T(vk )}.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. If T : V W is a linear transformation and {v1, ... , vk } is a linearly dependent set, then so is {T(v1), ... , T(vk )}.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. If T : R22 P6, then is it impossible for T to be onto.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. If T : P4 R6, then it is impossible for T to be one-to-one.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. Let T : V W be a linear transformation and w a nonzero vector in W. Then the set of all v in V such that T(v) = w forms a subspace.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. For every pair of vector spaces V and W, it is always possible to define a linear transformation T : V W.

Let T : R5 C[0, 1] be a linear transformation, and suppose that dim(ker(T)) = 2. What is dim(range(T))?

Let T : R43 P be a one-to-one linear transformation. What is dim(range(T))?

. Prove that if T : V W is a linear transformation with T(v1) = T(v2), then v1 v2 is in ker(T).

Prove that if T : V W is an onto and one-to-one linear transformation, and both V and W are of finite dimension, then dim(V) = dim(W).

Suppose that T1 : V W and T2 : V W are both linear transformations. Prove that T1 +T2 is also a linear transformation from V to W.

Prove the extended version of Theorem 9.2: If T : V W is a linear transformation, then T(c1v1 + c2v2 ++ cmvm) = c1T(v1) + c2T(v2) ++ cmT(vm), where v1, ... , vm are in V and c1, ... , cm are real scalars

Prove that if T : V W is a linear transformation, then T(0V ) = 0W.

Prove that if T : V W is a linear transformation, then for any v in V we have T(v) = T(v).

Prove part of Theorem 9.3: Let T : V W. Then ker(T) is a subspace of V.

Prove Theorem 9.2: T : V W is a linear transformation if and only if T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all vectors v1 and v2 in V and real scalars c1 and c2.

Prove Theorem 9.5: Let T : V W be a linear transformation. Then T is one-to-one if and only if ker(T) = {0V }. (HINT: Theorem 3.5 is similar.)

Suppose that T1 : V W and T2 : W Y are both linear transformations. Prove that T2  T1(v)  is also a linear transformation from V to Y.

Prove a partial converse of Theorem 9.6: Let T : V W be a one-to-one linear transformation, with V = {v1, ... , vm} a subset of V, and W = {w1, ... , wm} a subset of W. Suppose that T(vi) = wi for i = 1, ... , m. If V is linearly independent, then so is W.

Prove Theorem 9.8, but with the condition that V and W are finite dimensional removed.

(Calculus required) For Exercises 6368, some basic knowledge of calculus is required. Let C1(a, b) denote the set of functions that are continuously differentiable on the interval (a, b). Prove that T : C1(a, b) C(a, b) given by T( f ) = f  (x) is a linear transformation.

(Calculus required) For Exercises 6368, some basic knowledge of calculus is required. Let T : C[a, b] R be given by T( f ) =  b a f (x) dx. Prove that T is a linear transformation.

(Calculus required) For Exercises 6368, some basic knowledge of calculus is required. Determine if T : P4 P2 with T(p) = p(x) is a linear transformation.

(Calculus required) For Exercises 6368, some basic knowledge of calculus is required. Determine if T : P3 R with T(p) =  b a xp(x) dx is a linear transformation.

(Calculus required) For Exercises 6368, some basic knowledge of calculus is required. . Determine if T : P4 P5 with T(p) =  x2 p(x)  is a linear transformation.

(Calculus required) For Exercises 6368, some basic knowledge of calculus is required. Determine if T : P3 R with T(p) =  b a e x x2 p(x) dx is a linear transformation.