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**2-1 Linear Transformations,Null Spaces, and Ranges**
For Exercises 2 through 6, prove that T is a linear transformation, and find bases for both N(T) and R(T). Then compute the nullity and rank of T, and verify the dimension theorem. Finally, use the appropriate theorems in this section to determine whether T is one-to-one or onto.

2.$T: R^{3} → R^{2}$ defined by $T(a_{1}, a_{2}, a_{3})=(a_{1 }− a_{2}, 2a_{3})$.

3.$T: R^{2} → R^{3}$ defined by $T(a_{1}, a_{2})=(a_{1} + a_{2}, 0, 2a_{1} − a_{2})$.

10.Suppose that $T: R^{2} → R^{2}$ is linear, $T(1, 0) = (1, 4)$, and $T(1, 1) = (2, 5)$.
What is $T(2, 3)$? Is $T$ one-to-one?

13.Let $V$ and $W$ be vector spaces, let $T: V → W$ be linear, and let
{$w_{1}, w_{2},... ,w_{k}$} be a linearly independent subset of $R(T)$. Prove that if S = {$v_{1}, v_{2},... ,v_{k}$} is chosen so that $T(v_{i}) = w_{i}$ for i = 1, 2,... ,k, then S is linearly independent.

14.Let $V$ and $W$ be vector spaces and $T: V → W$ be linear.
(a.) Prove that $T$ is one-to-one if and only if $T$ carries linearly independent subsets of $V$ onto linearly independent subsets of $W$.
(b.) Suppose that $T$ is one-to-one and that $S$ is a subset of $V$. Prove that $S$ is linearly independent if and only if $T(S)$ is linearly independent.
(c.) Suppose β = {$v_{1}, v_{2},...,v_{n}$} is a basis for $V$ and $T$ is one-to-one and onto. Prove that$\enspace T(β)=${$T(v_{1}),T(v_{2}),...,T(v_{n})$} is a basis for $W$.

17.Let $V$ and $W$ be finite-dimensional vector spaces and $T: V → W$ be linear.
(a) Prove that if $dim(V) < dim(W)$, then $T$ cannot be onto.
(b) Prove that if $dim(V) > dim(W)$, then $T$ cannot be one-to-one.

20.Let $V$ and $W$ be vector spaces with subspaces $V_{1}$ and $W_{1}$, respectively.
If $T: V → W$ is linear, prove that $T(V_{1})$ is a subspace of $W$ and that {$x ∈ V: T(x) ∈ W_{1}$} is a subspace of $V$.

37.A function $T: V → W$ between vector spaces $V$ and $W$ is called $additive$ if $T(x + y) = T(x) + T(y)$ for all $x, y ∈ V$. Prove that if $V$ and $W$ are vector spaces over the field of rational numbers, then any additive function from $V$ into $W$ is a linear transformation.

38.Let $T: C → C$ be the function defined by $T(z) = \overline z$. Prove that $T$ is additive (as defined in Exercise 37) but not linear.

**2.2 THE MATRIX REPRESENTATION OF A LINEAR
TRANSFORMATION**

2.Let β and γ be the standard ordered bases for $R^n$ and $R^m$, respectively.
For each linear transformation $T: R^n → R^m$, compute $[T]^γ_{β}$.
(b) $T: R^3 → R^2$ defined by $T(a_{1}, a_{2}, a_{3}) = (2a_{1} + 3a_{2} − a_{3}, a_{1} + a_{3})$.

4.Define
$T: M_{2×2}(R) → P_{2}(R)$ by $T
\begin{pmatrix}{2}
a&b\\
c&d
\end{pmatrix}
= (a + b) + (2d)x + bx^2$.
Let
$β=
\begin{Bmatrix}
\begin{pmatrix}
1&0\\
0&0
\end{pmatrix}&
\begin{pmatrix}
0&1\\
0&0
\end{pmatrix}&
\begin{pmatrix}
0&1\\
0&0
\end{pmatrix}&
\begin{pmatrix}
0&0\\
0&1
\end{pmatrix}
\end{Bmatrix}$ and $γ = \begin{Bmatrix} 1,&x,&x^2 \end{Bmatrix}
.$

5.Let
$α =\begin{Bmatrix}
\begin{pmatrix}
1&0\\
0&0
\end{pmatrix}&
\begin{pmatrix}
0&1\\
0&0
\end{pmatrix}&
\begin{pmatrix}
0&1\\
0&0
\end{pmatrix}&
\begin{pmatrix}
0&0\\
0&1
\end{pmatrix}
\end{Bmatrix}$
β = {$1, x, x^{2}$},
and γ = {1}.

(b) Define
$T: P_{2}(R) → M_{2×2}(R)$ by $T(f(x)) =
\begin{pmatrix}
f'(0)&2f(1)\\
0&f''(3)
\end{pmatrix}.$

(d) Define $T: P_{2}(R) → R \enspace by \enspace T(f(x)) = f(2).$ Compute $[T]^γ_{β}$.

(f) If $f(x)=3 − 6x + x^2$, compute $[f(x)]_{β}$.

8.Let $V$ be an n-dimensional vector space with an ordered basis β. Define
$T: V → F^n$ by $T(x)=[x]_{β}$. Prove that $T$ is linear.

9.Let $V$ be the vector space of complex numbers over the field R. Define
$T: V → V$ by $T(z) = \overline z$, where z is the complex conjugate of z. Prove that $T$ is linear, and compute $[T]_{β}$, where β = {1, i}. (Recall by Exercise 38 of Section 2.1 that $T$ is not linear if $V$ is regarded as a vector space over the field $C$.)

13.Let $V$ and $W$ be vector spaces, and let $T$ and $U$ be nonzero linear
transformations from $V$ into $W$. If $R(T) ∩ R(U) =${$0$}, prove that ${T,U}$ is a linearly independent subset of $L(V, W)$.

14.Let $V = P(R)$, and for j ≥ 1 define $T_{j} (f(x)) = f(j)(x)$, where $f(j)(x)$ is the jth derivative of $f(x)$. Prove that the set {$T_{1},T_{2},...,T_{n}$} is a linearly independent subset of $L(V)$ for any positive integer n.

**2.3 COMPOSITION OF LINEAR TRANSFORMATIONS
AND MATRIX MULTIPLICATION**

2.(b) Let
$A = \begin{pmatrix}
2&5\\
-3&1\\
4&2\end{pmatrix}$,$B= \begin{pmatrix}
3&-2&0\\
1&-1&4\\
5&5&3\end{pmatrix}$, and $C=(4 \enspace 0 \enspace 3)$.
Compute $A^t$, $A^tB$, $BC^t$, $CB$, and $CA$

3.Let $g(x)=3+ x$. Let $T: P_{2}(R) → P_{2}(R)$ and $U: P_{2}(R) → R^3$ be the linear transformations respectively defined by
$T(f(x)) = f(x)g(x)+2f(x)$ and $U(a + bx + cx^2)=(a + b, c, a − b)$.
Let β and γ be the standard ordered bases of $P_{2}(R)$ and $R^3$, respectively.

4.For each of the following parts, let $T$ be the linear transformation defined in the corresponding part of Exercise 5 of Section 2.2. Use Theorem 2.14 to compute the following vectors:
(b) $[T(f(x))]_{α}$, where $f(x)=4 − 6x + 3x^2$.

11.Let $V$ beavector space, and let $T: V → V$ be linear. Prove that $T^2 = T_{0}$ if and only if $R(T) ⊆ N(T)$.

13.Let A and B be n × n matrices. Recall that the trace of A is defined by
$tr(A)=\sum_{i=1}^nA_{ii}$
Prove that $tr(AB) = tr(BA)$ and $tr(A) = tr(At
)$.

**2.4 INVERTIBILITY AND ISOMORPHISMS**

2.For each of the following linear transformations $T$, determine whether $T$ is invertible and justify your answer.
(b) $T: R^2 → R^3$ defined by $T(a_{1}, a_{2}) = (3a_{1} − a_{2}, a_{2}, 4a_1)$.

3.Which of the following pairs of vector spaces are isomorphic? Justify your answers.
(a) $F^3$ and $P_3(F)$.

4.Let A and B be n × n invertible matrices. Prove that AB is invertible
and $(AB)^{−1} = B^{−1}A^{−1}$.

5.Let A be invertible. Prove that At is invertible and $(A^t
)^{−1} = (A^{−1})^t$.

7.Let A be an n × n matrix.
(a) Suppose that $A^2 = O$. Prove that A is not invertible.
(b) Suppose that $AB = O$ for some nonzero n × n matrix B. Could A be invertible? Explain.

10.Let A and B be n × n matrices such that $AB = I_{n}$.
(a) Use Exercise 9 to conclude that A and B are invertible.
(b) Prove $A = B^{−1}$ (and hence $B = A^{−1}$). (We are, in effect, saying that for square matrices, a “one-sided” inverse is a “two-sided” inverse.)
(c) State and prove analogous results for linear transformations defined on finite-dimensional vector spaces.

13.Let ∼ mean “is isomorphic to.” Prove that ∼ is an equivalence relation on the class of vector spaces over $F$.

15.Let $V$ and $W$ be finite-dimensional vector spaces, and let $T: V → W$ be a linear transformation. Suppose that β is a basis for $V$. Prove that $T$ is an isomorphism if and only if $T(β)$ is a basis for $W$.

17.Let $V$ and $W$ be finite-dimensional vector spaces and $T: V → W$ be an isomorphism. Let $V_0$ be a subspace of $V$.
(a) Prove that $T(V_0)$ is a subspace of $W$.
(b) Prove that $dim(V_0) = dim(T(V_0))$.

20.Let $T: V → W$ be a linear transformation from an n-dimensional vector space $V$ to an m-dimensional vector space $W$. Let β and γ be ordered bases for $V$ and $W$, respectively.
Prove that $rank(T) = rank(L_A)$ and that $nullity(T) = nullity(L_A)$, where A = $[T]^γ_β$. Hint: Apply Exercise 17 to Figure 2.2.

**2.5 THE CHANGE OF COORDINATE MATRIX**

2.For each of the following pairs of ordered bases β and β' for $R^2$, find the change of coordinate matrix that changes β'-coordinates into βcoordinates.
(b) β = {(−1, 3),(2, −1)} and β = {(0, 10),(5, 0)}

3.For each of the following pairs of ordered bases β and β' for $P_2(R)$, find the change of coordinate matrix that changes β'-coordinates into β-coordinates.
(c) β = {$2x^2 − x, 3x^2 + 1, x^2$} and β = {$1, x, x^2$}

4.Let $T$ be the linear operator on $R^2$ defined by
$T\begin{pmatrix}
a\\
b\end{pmatrix}=
\begin{pmatrix}
2a+b\\
a-3b\end{pmatrix}$
let β be the standard ordered basis for $R^2$, and let
$β'=\begin{Bmatrix}
\begin{pmatrix}
1\\
1\end{pmatrix}&\begin{pmatrix}
1\\
2\end{pmatrix}\end{Bmatrix}$
Use Theorem 2.23 and the fact that
$\begin{pmatrix}
1&1\\
1&2\end{pmatrix}^{-1}=\begin{pmatrix}
2&-1\\-1&1\end{pmatrix}$
to find $[T]_{β'}$ .

8.Prove the following generalization of Theorem 2.23. Let $T: V → W$ be a linear transformation from a finite-dimensional vector space $V$ to a finite-dimensional vector space $W$. Let β and β be ordered bases for $V$, and let γ and γ be ordered bases for $W$. Then $[T]^{γ'}_{β'} = P ^{−1}[T]^γ_βQ$,where Q is the matrix that changes β'-coordinates into β-coordinates and P is the matrix that changes γ'-coordinates into γ-coordinates.

9.Prove that “is similar to” is an equivalence relation on $M_{n×n}(F)$.

Let $V$ be a finite-dimensional vector space with ordered bases α, β,and γ.
(a) Prove that if Q and R are the change of coordinate matrices that change α-coordinates into β-coordinates and β-coordinates into γ-coordinates, respectively, then RQ is the change of coordinate matrix that changes α-coordinates into γ-coordinates.
(b) Prove that if Q changes α-coordinates into β-coordinates, then $Q^{−1}$ changes β-coordinates into α-coordinates.

Let $V$ be a finite-dimensional vector space over a field $F$, and let β ={$x_1, x_2,...,x_n$} be an ordered basis for $V$. Let Q be an n × n invertible matrix with entries from $F$. Define
$x'_j=\sum_{i=1}^nQ_{ij}x_j$ for 1≤ j ≤ n,
and set β = {$x'_1, x'_2,...,x'_n$}. Prove that β is a basis for V and hence that Q is the change of coordinate matrix changing β-coordinates into β-coordinates.

**4.1 DETERMINANTS OF ORDER 2**

2.Compute the determinants of the following matrices in $M_{2×2}(R)$.
$(a)\enspace \begin{pmatrix}
6&-3\\
2&4\end{pmatrix}$

3.Compute the determinants of the following matrices in $M_{2×2}(C)$.
$(a)\enspace \begin{pmatrix}
-1+i&1-4i\\
3+2i&2-3i\end{pmatrix}$

4.For each of the following pairs of vectors u and v in R^2, compute the area of the parallelogram determined by u and v.
(c) $u = (4, −1)$ and $v = (−6, −2)$

5.Prove that if B is the matrix obtained by interchanging the rows of a 2 × 2 matrix A, then $det(B) = − det(A)$.

6.Prove that if the two columns of $A ∈ M_{2×2}(F)$ are identical, then det(A) = 0.

7.Prove that $det(A^t) = det(A)$ for any $A ∈ M_{2×2}(F)$.

8.Prove that if $A ∈ M_{2×2}(F)$ is upper triangular, then $det(A)$ equals the product of the diagonal entries of A.

**4.2 DETERMINANTS OF ORDER n**

2.Find the value of k that satisfies the following equation:
$det\begin{pmatrix}
3a_1&3a_2&3a_3\\
3b_1&3b_2&3b_3\\
3c_1&3c_2&3c_3\end{pmatrix}=kdet
\begin{pmatrix}
a_1&a_2&a_3\\
b_1&b_2&b_3\\
c_1&c_2&c_3\end{pmatrix}$

In Exercises 5–12, evaluate the determinant of the given matrix by cofactor
expansion along the indicated row.
$5.\enspace \begin{pmatrix}
0&1&2\\
-1&0&-3\\
-1&3&0\end{pmatrix}$
along the first row

$9.\enspace \begin{pmatrix}
0&1+i&2\\
-2i&0&1-i\\
3&4i&0\end{pmatrix}$
along the third row

In Exercises 13–22, evaluate the determinant of the given matrix by any legitimate method.
$15.\enspace \begin{pmatrix}
1&2&3\\
4&5&6\\
7&8&9\end{pmatrix}$

$21.\enspace \begin{pmatrix}
1&0&-2&3\\
-3&1&1&2\\
0&4&-1&1\\
2&3&0&1\end{pmatrix}$

23.Prove that the determinant of an upper triangular matrix is the product of its diagonal entries.

25.Prove that $det(kA) = k^n det(A)$ for any A ∈ M_{n×n}(F).

27.Prove that if $A ∈ M_{n×n}(F)$ has two identical columns, then $det(A) = 0$.

29.Prove that if E is an elementary matrix, then det(E^t) = det(E).

**4.3 PROPERTIES OF DETERMINANTS**

In Exercises 2–7, use Cramer’s rule to solve the given system of linear equations.
$2.\enspace \begin{matrix}
a_{11}x_1 + a_{12}x_2 = b_1\\
a_{21}x_1 + a_{22}x_2 = b_2\\
where\enspace a_{11}a{22} − a{12}a{21} \neq 0\end{matrix}$

$6.\enspace \begin{matrix}
x_1 − x_2 + 4x_3 = −2\\
−8x_1 + 3x_2 + x_3 = 0\\
2x_1 − x_2 + x_3 = 6\end{matrix}$

9.Prove that an upper triangular n × n matrix is invertible if and only if all its diagonal entries are nonzero.

15.Prove that if $A, B ∈ M_{n×n}(F)$ are similar, then $det(A) = det(B)$.

20.Suppose that M ∈ M_{n×n}(F) can be written in the form
$M=\begin{pmatrix}
A&B\\
O&I\end{pmatrix}$,
where A is a square matrix. Prove that $det(M) = det(A)$.

21.Prove that if $M ∈ M_{n×n}(F)$ can be written in the form
$M=\begin{pmatrix}
A&B\\
O&C\end{pmatrix}$,
where A and C are square matrices, then $det(M) = det(A)· det(C)$.

**4.4 SUMMARY—IMPORTANT FACTS ABOUT DETERMINANTS**

Evaluate the determinant of the following 2 × 2 matrices.
$(a)\enspace \begin{pmatrix}
4&-5\\
2&3\end{pmatrix}$

3.Evaluate the determinant of the following matrices in the manner indicated.
$(c)\enspace \begin{pmatrix}
0&1&2\\
-1&0&-3\\
2&3&0\end{pmatrix}$
along the second column

$(e)\enspace \begin{pmatrix}
0&1+i&2\\
-2i&0&1-i\\
3&4i&0\end{pmatrix}$
along the third row

4.Evaluate the determinant of the following matrices by any legitimate method.
$(a)\enspace \begin{pmatrix}
1&2&3\\
4&5&6\\
7&8&9\end{pmatrix}$

$(e)\enspace \begin{pmatrix}
i&2&-1\\
3&1+i&2\\
-2i&1&4-i\end{pmatrix}$