Section 1
Definition of a Linear Transformation
Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation.$T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ defined by $$ T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}+3 x_{2}+x_{3}, x_{1}-x_{2}\right) $$.
Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation.$T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ defined by $$ T\left(x_{1}, x_{2}\right)=\left(x_{1}+2 x_{2}, 2 x_{1}-x_{2}\right) $$.
Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation.$T: C^{2}(I) \rightarrow C^{0}(I)$ defined by $$ T(y)=y^{\prime \prime}+a_{1} y^{\prime}+a_{2} y $$ where $a_{1}$ and $a_{2}$ are functions defined on $I$.
Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation.$T: C^{2}(I) \rightarrow C^{0}(I)$ defined by $$ T(y)=y^{\prime \prime}-16 y $$.
Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation.$T: M_{n}(\mathbb{R}) \rightarrow M_{n}(\mathbb{R})$ defined by $$ T(A)=A B-B A $$ where $B$ is a fixed $n \times n$ matrix.
Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation.$T: C^{0}[a, b] \rightarrow \mathbb{R}$ defined by $$ T(f)=\int_{a}^{b} f(x) d x $$.
Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation.$T: M_{n}(\mathbb{R}) \rightarrow \mathbb{R}$ defined by $T(A)=\operatorname{tr}(A),$ where $\operatorname{tr}(A)$ denotes the trace of $A$.
Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation.$S: M_{n}(\mathbb{R}) \rightarrow M_{n}(\mathbb{R})$ defined by $$ S(A)=A+A^{T} $$.
Show that the given mapping is a nonlinear transformation.$$\begin{aligned} &T: P_{2}(\mathbb{R}) \rightarrow \mathbb{R} \text { defined by }\\ &T\left(a+b x+c x^{2}\right)=a+b+c+1 \end{aligned}$$.
Show that the given mapping is a nonlinear transformation.$$T: M_{2}(\mathbb{R}) \rightarrow M_{2}(\mathbb{R}) \text { defined by } T(A)=A^{2}$$.
Show that the given mapping is a nonlinear transformation.$$T: C^{0}[a, b] \rightarrow C^{0}[a, b] \text { defined by } T(f(x))=x$$.
Show that the given mapping is a nonlinear transformation.$T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ defined by $$ T\left(x_{1}, x_{2}\right)=\left(x_{1}+x_{2}, 2\right) $$.
Show that the given mapping is a nonlinear transformation.$$ T\left(x_{1}, x_{2}\right)=\left(x_{1}+x_{2}, 2\right) $$$T: M_{2}(\mathbb{R}) \rightarrow \mathbb{R}$ defined by $$ T(A)=\operatorname{det}(A) $$.
For Problems $14-18$, determine the matrix of the given transformation$$T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$$$T\left(x_{1}, x_{2}\right)=\left(3 x_{1}-2 x_{2}, x_{1}+5 x_{2}\right)$
Determine the matrix of the given transformation $$T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$$.$$T\left(x_{1}, x_{2}\right)=\left(x_{1}+3 x_{2}, 2 x_{1}-7 x_{2}, x_{1}\right)$$.
Determine the matrix of the given transformation $$T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$$.$$T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}-x_{2}+x_{3}, x_{3}-x_{1}\right)$$.
Determine the matrix of the given transformation $$T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$$.$$T\left(x_{1}, x_{2}, x_{3}\right)=x_{1}+5 x_{2}-3 x_{3}$$.
Determine the matrix of the given transformation $$T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$$.$$T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{3}-x_{1},-x_{1}, 3 x_{1}+2 x_{3}, 0\right)$$.
Determine the linear transformation $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ that has the given matrix.$$A=\left[\begin{array}{rr} 1 & 3 \\ -4 & 7 \end{array}\right]$$.
Determine the linear transformation $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ that has the given matrix.$$A=\left[\begin{array}{rrr} 2 & -1 & 5 \\ 3 & 1 & -2 \end{array}\right]$$.
Determine the linear transformation $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ that has the given matrix.$$A=\left[\begin{array}{rrr} 2 & 2 & -3 \\ 4 & -1 & 2 \\ 5 & 7 & -8 \end{array}\right]$$.
Determine the linear transformation $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ that has the given matrix.$$A=\left[\begin{array}{r} -3 \\ -2 \\ 0 \\ 1 \end{array}\right]$$.
Determine the linear transformation $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ that has the given matrix.$$A=\left[\begin{array}{lllll} 1 & -4 & -6 & 0 & 2 \end{array}\right]$$.
Let $V$ be a real inner product space, and let u be a fixed (nonzero) vector in $V .$ Define $T: V \rightarrow \mathbb{R}$ by $$T(\mathbf{v})=\left(\left\langle\mathbf{u}_{1}, \mathbf{v}\right\rangle,\left\langle\mathbf{u}_{2}, \mathbf{v}\right\rangle\right)$$. Use properties of the inner product to show that $T$ is a linear transformation.
Let $V$ be a real inner product space, and let $\mathbf{u}_{1}$ and $\mathbf{u}_{2}$ be fixed (nonzero) vectors in $V .$ Define $T: V \rightarrow \mathbb{R}^{2}$ by $$T(\mathbf{v})=\left(\left\langle\mathbf{u}_{1}, \mathbf{v}\right\rangle,\left\langle\mathbf{u}_{2}, \mathbf{v}\right\rangle\right)$$. Use properties of the inner product to show that $T$ is a linear transformation.
(a) Let $\mathbf{v}_{1}=(1,1)$ and $\mathbf{v}_{2}=(1,-1) .$ Show that $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$ is a basis for $\mathbb{R}^{2}.$(b) Let $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be the linear transformation satisfying $$ T\left(\mathbf{v}_{1}\right)=(2,3), \quad T\left(\mathbf{v}_{2}\right)=(-1,1) $$ where $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ are the basis vectors given in (a). Find $T\left(x_{1}, x_{2}\right)$ for an arbitrary vector $\left(x_{1}, x_{2}\right)$ in $\mathbb{R}^{2} .$ What is $T(4,-2) ?$
Assume that $T$ defines a linear transformation and use the given information to find the matrixof $T.$$T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{4}$ such that $T(-1,1)=(1,0,-2,2)$ and $T(1,2)=(-3,1,1,1).$
Assume that $T$ defines a linear transformation and use the given information to find the matrixof $T.$$T: \mathbb{R}^{4} \rightarrow \mathbb{R}^{2}$ such that $T(1,0,0,0)=(3,-2)$ $T(1,1,0,0)=(5,1), T(1,1,1,0)=(-1,0),$ and $T(1,1,1,1)=(2,2)$.
Assume that $T$ defines a linear transformation and use the given information to find the matrixof $T.$$$\begin{aligned} &T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} \text { such that } T(1,2,0)=(2,-1,1),\\ &T(0,1,1)=(3,-1,-1) \text { and } T(0,2,3)=(6,-5,4) \end{aligned}$$.
Assume that $T$ defines a linear transformation and use the given information to find the matrixof $T.$$$\begin{aligned} &T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{4} \text { such that } T(0,-1,4)=(2,5,-2,1)\\ &T(0,3,3)=(-1,0,0,5), \text { and } T(4,4,-1)=\\ &(-3,1,1,3) \end{aligned}$$.
Let $T: P_{2}(\mathbb{R}) \rightarrow P_{2}(\mathbb{R})$ be the linear transformation satisfying $$T(1)=x+1, \quad T(x)=x^{2}-1, \quad T\left(x^{2}\right)=3 x+2$$. Determine $T\left(a x^{2}+b x+c\right),$ where $a, b,$ and $c$ are arbitrary real numbers.
Let $T: V \rightarrow V$ be a linear transformation, and suppose that $$ \begin{aligned} T\left(2 \mathbf{v}_{1}+3 \mathbf{v}_{2}\right) &=\mathbf{v}_{1}+\mathbf{v}_{2} \\ T\left(\mathbf{v}_{1}+\mathbf{v}_{2}\right) &=3 \mathbf{v}_{1}-\mathbf{v}_{2}. \end{aligned} $$ Find $T\left(\mathbf{v}_{1}\right)$ and $T\left(\mathbf{v}_{2}\right)$.
Let $T: P_{2}(\mathbb{R}) \rightarrow P_{2}(\mathbb{R})$ be the linear transformation satisfying: $$\begin{array}{c} T\left(x^{2}-1\right)=x^{2}+x-3, \quad T(2 x)=4 x ,\\ T(3 x+2)=2(x+3). \end{array}$$ Find $T(1), T(x), T\left(x^{2}\right),$ and hence show that $$ T\left(a x^{2}+b x+c\right)=a x^{2}-(a-2 b+2 c) x+3 c, $$ where $a, b,$ and $c$ are arbitrary real numbers.
Let $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$ be a basis for the vector space $V .$ If $T: V \rightarrow V$ is the linear transformation satisfying Let $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$ be a basis for the vector space $V .$ If $T: V \rightarrow V$ is the linear transformation satisfying, find $T(\mathbf{v})$ for an arbitrary vector in $V.$
Let $T: V \rightarrow W$ and $S: V \rightarrow W$ be linear transformations, and assume that $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\}$ spans $V$ Prove that if $T\left(\mathbf{v}_{i}\right)=S\left(\mathbf{v}_{i}\right)$ for each $i=1,2, \ldots, k$ then $T=S ;$ that is, $T(\mathbf{v})=S(\mathbf{v})$ for each $\mathbf{v} \in V.$
Let $V$ be a vector space with basis $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\}$ and suppose $T: V \rightarrow W$ is a linear transformation such that $T\left(\mathbf{v}_{i}\right)=\mathbf{0}$ for each $i=1,2, \ldots, k .$ Prove that $T$ is the zero transformation; that is, $T(\mathbf{v})=\mathbf{0}$ for each $\mathbf{v} \in V.$
Let $T_{1}: V \rightarrow W$ and $T_{2}: V \rightarrow W$ be linear transformations, and let $c$ be a scalar. We define the sum $T_{1}+T_{2}$ and the scalar product $c T_{1}$ by $$ \left(T_{1}+T_{2}\right)(\mathbf{v})=T_{1}(\mathbf{v})+T_{2}(\mathbf{v}) $$ and $$ \left(c T_{1}\right)(\mathbf{v})=c T_{1}(\mathbf{v}) $$ for all $\mathbf{v} \in V .$ The remaining problems in this section consider the properties of these mappings.Verify that $T_{1}+T_{2}$ and $c T_{1}$ are linear transformations.
Let $T_{1}: V \rightarrow W$ and $T_{2}: V \rightarrow W$ be linear transformations, and let $c$ be a scalar. We define the sum $T_{1}+T_{2}$ and the scalar product $c T_{1}$ by $$ \left(T_{1}+T_{2}\right)(\mathbf{v})=T_{1}(\mathbf{v})+T_{2}(\mathbf{v}) $$ and $$ \left(c T_{1}\right)(\mathbf{v})=c T_{1}(\mathbf{v}) $$ for all $\mathbf{v} \in V .$ The remaining problems in this section consider the properties of these mappings.Let $T_{1}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ and $T_{2}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be the lineartransformations with matrices $$ A=\left[\begin{array}{rr} 3 & 1 \\ -1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 5 \\ 3 & -4. \end{array}\right] $$ Find $T_{1}+T_{2}$ and $c T_{1}.$
Let $T_{1}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ and $T_{2}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be the linear transformations with matrices $A$ and $B$ respectively. Show that $T_{1}+T_{2}$ and $c T_{1}$ are the linear transformations with matrices $A+B$ and $c A$ respectively.
Let $V$ and $W$ be vector spaces, and let $L(V, W)$ denote the set of all linear transformations from $V$ into $W .$ Verify that $L(V, W)$ together with the operations of addition and scalar multiplication just defined for linear transformations is a vector space.