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## Linear Transformations

Definition. A function $$T: V \to W$$ from a subspace $$V$$ of $$\mathbb R^n$$ to a subspace $$W$$ of $$\mathbb R^m$$ is called a linear transformation if

1. $$T(\overrightarrow{u}+\overrightarrow{v})= T(\overrightarrow{u})+T(\overrightarrow{v})$$ for all $$\overrightarrow{u}, \overrightarrow{v} \in V$$ and

2. $$T(c\overrightarrow{v})=cT(\overrightarrow{v})$$ for for all $$\overrightarrow{v}\in V$$ and all scalars $$c \in \mathbb R$$.

In short, a function $$T: V \to W$$ is a linear transformation if it preserves the linearity among vectors:
$$T(c\overrightarrow{u}+d\overrightarrow{v})= cT(\overrightarrow{u})+dT(\overrightarrow{v})$$ for all $$\overrightarrow{u}, \overrightarrow{v} \in V$$ and all scalars $$c,d \in \mathbb R$$.

Example.

1. The projection $$T:\mathbb R^3 \to \mathbb R^3$$ of $$\mathbb R^3$$ onto the $$xy$$-plane in $$\mathbb R^3$$ is defined by $T \left( \left[\begin{array}{c}x_1\\x_2\\x_3 \end{array} \right] \right) = \left[\begin{array}{c}x_1\\x_2\\0 \end{array} \right] \text{ for all } \overrightarrow{x}=\left[\begin{array}{c}x_1\\x_2\\x_3 \end{array} \right] \in \mathbb R^3.$ Sometimes it is simply denoted by $$T(x_1,x_2,x_3)=(x_1,x_2,0)$$ in terms of row vectors. To show it is a linear transformation, let $$\overrightarrow{x}=(x_1,x_2,x_3)$$ and $$\overrightarrow{y}=(y_1,y_2,y_3)$$ in $$\mathbb R^3$$ and $$c,d\in \mathbb R$$. Then \begin{align*} T(c\overrightarrow{x}+d\overrightarrow{y}) &= T(cx_1+dy_1,cx_2+dy_2,cx_3+dy_3)\\ &= (cx_1+dy_1,cx_2+dy_2,0)\\ &= (cx_1,cx_2,0)+(dy_1,dy_2,0)\\ &=cT(\overrightarrow{x})+dT(\overrightarrow{y}). \end{align*}

2. For the matrix $$A=\left[\begin{array}{rr}1&2\\0&1 \end{array}\right]$$, define the shear transformation $$T:\mathbb R^2\to \mathbb R^2$$ by $$T(\overrightarrow{x})=A\overrightarrow{x}$$. Let $$\overrightarrow{x}, \overrightarrow{y}\in \mathbb R^2$$ and $$c,d\in \mathbb R$$. Then $T(c\overrightarrow{x}+d\overrightarrow{y})=A(c\overrightarrow{x}+d\overrightarrow{y}) = cA \overrightarrow{x}+dA \overrightarrow{y}= cT(\overrightarrow{x})+dT(\overrightarrow{y}).$ Thus $$T$$ is a linear transformation which transforms the square formed by $$(0,0)$$,$$(1,0)$$,$$(1,1)$$,$$(0,1)$$ to the parallelogram formed by $$(0,0),(1,0),(3,1),(2,1)$$.

Definition. A matrix transformation is the linear transformation $$T:\mathbb R^n\to \mathbb R^m$$ defined by $$T(\overrightarrow{x})=A\overrightarrow{x}$$ for some $$m\times n$$ matrix $$A$$. It is denoted by $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$.

From the definition of a linear transformation we have the following properties.
Property For a linear transformation $$T: V \to W$$ where $$V\leq \mathbb R^n$$ and $$W\leq \mathbb R^m$$,

1. $$T(\overrightarrow{0_n})=\overrightarrow{0_m}$$ and

2. for all $$\overrightarrow{v_1}, \ldots,\overrightarrow{v_k} \in V$$ and all $$c_1,\ldots,c_k \in \mathbb R$$, $T(c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}+\cdots+c_k\overrightarrow{v_k}) = c_1 T(\overrightarrow{v_1})+c_2 T(\overrightarrow{v_2})+\cdots+c_k T(\overrightarrow{v_k}).$

Example. Consider the function $$T:\mathbb R^3 \to \mathbb R^3$$ defined by $$T(x_1,x_2,x_3)=(x_1,x_2,5)$$. Since $$T(0,0,0)=(0,0,5) \neq (0,0,0)$$, $$T$$ is not a linear transformation.

Theorem. For a linear transformation $$T:\mathbb R^n \to \mathbb R^m$$, there exists a unique $$m\times n$$ matrix $$A$$, called the standard matrix of $$T$$, for which $T(\overrightarrow{x})=A\overrightarrow{x} \text{ for all } \overrightarrow{x}\in \mathbb R^n.$ Moreover, $$A=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) \:\cdots T(\overrightarrow{e_n})]$$ where $$\overrightarrow{e_i}$$ is the $$i$$th column of $$I_n$$.

Let $$\overrightarrow{x}=[x_1,x_2,\ldots,x_n]^T \in \mathbb R^n$$. We can write $$\overrightarrow{x}=x_1\overrightarrow{e_1}+x_2\overrightarrow{e_2}+\cdots+x_n\overrightarrow{e_n}$$. Then \begin{align*} T(\overrightarrow{x})=T(x_1\overrightarrow{e_1}+x_2\overrightarrow{e_2}+\cdots+x_n\overrightarrow{e_n}) &=x_1 T(\overrightarrow{e_1})+x_2 T(\overrightarrow{e_2})+\cdots+x_n T(\overrightarrow{e_n})\\ &=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) \:\cdots T(\overrightarrow{e_n})] \left[\begin{array}{c}x_1\\x_2\\ \vdots\\x_n \end{array} \right]\\ &= A\overrightarrow{x}. \end{align*}

Example.

1. Use the standard matrix to find the rotation transformation $$T:\mathbb R^2 \to \mathbb R^2$$ that rotates each point of $$\mathbb R^2$$ about the origin through an angle $$\theta$$ counterclockwise.

Solution. By trigonometry we have $T(\overrightarrow{e_1})=T\left(\left[\begin{array}{c}1\\0 \end{array} \right] \right)=\left[\begin{array}{c}\cos \theta \\ \sin\theta \end{array} \right] \text{ and } T(\overrightarrow{e_2})=T\left(\left[\begin{array}{c}0\\1 \end{array} \right] \right)=\left[\begin{array}{r} -\sin\theta\\ \cos \theta \end{array} \right].$ Then the standard matrix is $$A=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) ] =\left[\begin{array}{rr} \cos \theta& -\sin\theta\\ \sin\theta& \cos \theta \end{array} \right].$$ Thus $T(\overrightarrow{x})=A\overrightarrow{x}, \text{ i.e., } T\left( \left[\begin{array}{c}x_1\\x_2 \end{array} \right] \right) =\left[\begin{array}{c}x_1\cos \theta -x_2\sin\theta \\ x_1\sin\theta +x_2\cos \theta \end{array} \right] \text{ for all } \overrightarrow{x}\in \mathbb R^2.$

2. Consider the linear transformation $$T:\mathbb R^2 \to \mathbb R^3$$ defined by $T(x_1,x_2)=(x_1-x_2,2x_1+3x_2,4x_2).$ Note that $$T(\overrightarrow{e_1})=T(1,0)=(1,2,0)$$ and $$T(\overrightarrow{e_2})=T(0,1)=(-1,3,4)$$. The standard matrix of $$T$$ is $A=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) ] =\left[\begin{array}{rr} 1&-1\\ 2&3\\ 0&4 \end{array} \right].$

For any given linear transformation $$T:\mathbb R^n \to \mathbb R^m$$, the domain space is $$\mathbb R^n$$ and the codomain space is $$\mathbb R^m$$. We study a subspace of the domain space called Kernel or Null Space and a subspace of the codomain space called Image Space or Range.

Definition. The kernel or null space of a linear transformation $$T:\mathbb R^n \to \mathbb R^m$$, denoted by $$\ker (T)$$ or $$\ker T$$, is the following subspace of $$\mathbb R^n$$: $\ker T= \{\overrightarrow{x} \in \mathbb R^n \;|\; T(\overrightarrow{x})=\overrightarrow{0_m}\}.$ The nullity of $$T$$, denoted by $$\operatorname{nullity}(T)$$, is the dimension of $$\ker T$$, i.e., $\operatorname{nullity}(T)=\operatorname{dim}(\ker T).$
Remark. If $$A$$ is the standard matrix of a linear transformation $$T:\mathbb R^n \to \mathbb R^m$$, then $$\ker T=\operatorname{NS}(A)$$ and $$\operatorname{nullity}(T)=\operatorname{nullity}(A)$$.

Example. The linear transformation $$T:\mathbb R^3 \to \mathbb R^2$$ defined by $$T(x_1,x_2,x_3)=(x_1,x_2)$$ has the standard matrix $$A=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) \: T(\overrightarrow{e_3})] =\left[\begin{array}{rrr} 1&0&0\\ 0&1&0 \end{array} \right]$$. Note that $\ker T=\operatorname{NS}(A)=\operatorname{Span} \left\lbrace \left[\begin{array}{r} 0\\0\\1 \end{array} \right] \right\rbrace,$ and $$\operatorname{nullity}(T)=\operatorname{nullity}(A)=1$$.

Definition. The image space or range of a linear transformation $$T:\mathbb R^n \to \mathbb R^m$$, denoted by $$\operatorname{im} (T)$$ or $$\operatorname{im} T$$ or $$T(\mathbb R^n)$$, is the following subspace of $$\mathbb R^m$$: $\operatorname{im} T= \{T(\overrightarrow{x}) \;|\; \overrightarrow{x} \in \mathbb R^n\}.$ The rank of $$T$$, denoted by $$\operatorname{rank}(T)$$, is the dimension of $$\operatorname{im} T$$, i.e., $\operatorname{rank}(T)=\operatorname{dim}(\operatorname{im} T).$

Remark. If $$A$$ is the standard matrix of a linear transformation $$T:\mathbb R^n \to \mathbb R^m$$, then $$\operatorname{im} T=\operatorname{CS}\left(A\right)$$ and $$\operatorname{rank}(T)=\operatorname{rank}(A)$$.
Example. The linear transformation $$T:\mathbb R^2 \to \mathbb R^3$$ defined by $$T(x_1,x_2)=(x_1,x_2,0)$$ has the standard matrix $$A=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) ] =\left[\begin{array}{rr} 1&0\\ 0&1\\ 0&0 \end{array} \right]$$. Note that $\operatorname{im} T=\operatorname{CS}\left(A\right)=\operatorname{Span} \left\lbrace \left[\begin{array}{r} 1\\0\\0 \end{array} \right], \left[\begin{array}{r} 0\\1\\0 \end{array} \right] \right\rbrace,$ and $$\operatorname{rank}(T)=\operatorname{rank}(A)=2$$.

Theorem.(Rank-Nullity Theorem) For a linear transformation $$T:\mathbb R^n \to \mathbb R^m$$, $\operatorname{rank}(T)+\operatorname{nullity}(T)=n.$

Let $$A$$ be the $$m\times n$$ standard matrix of $$T$$. Then by the Rank-Nullity Theorem on $$A$$, $\operatorname{rank}(T)+\operatorname{nullity}(T)=\operatorname{rank}(A)+\operatorname{nullity}(A)=n.$

Example. The linear transformation $$T:\mathbb R^3 \to \mathbb R^2$$ defined by $$T(x_1,x_2,x_3)=(x_1,x_2)$$ has $$\operatorname{nullity}(T)=1$$ (see examples before). Then by the Rank-Nullity Theorem, $\operatorname{rank}(T)=3-\operatorname{nullity}(T)=2.$

Now we discuss two important types of linear transformation $$T:\mathbb R^n \to \mathbb R^m$$.
Definition. Let $$T:\mathbb R^n \to \mathbb R^m$$ be a linear transformation. $$T$$ is onto if each $$\overrightarrow{b}\in \mathbb R^m$$ has a pre-image $$\overrightarrow{x}$$ in $$\mathbb R^n$$ under $$T$$, i.e., $$T(\overrightarrow{x})=\overrightarrow{b}$$. $$T$$ is one-to-one if each $$\overrightarrow{b}\in \mathbb R^m$$ has at most one pre-image in $$\mathbb R^n$$ under $$T$$.

Example.

1. The linear transformation $$T:\mathbb R^3 \to \mathbb R^2$$ defined by $$T(x_1,x_2,x_3)=(x_1,x_2)$$ is onto because each $$(x_1,x_2)\in \mathbb R^2$$ has a pre-image $$(x_1,x_2,0)\in \mathbb R^3$$ under $$T$$. But $$T$$ is not one-to-one because $$T(0,0,0)=T(0,0,1)=(0,0)$$, i.e., $$(0,0)$$ has two distinct pre-images $$(0,0,0)$$ and $$(0,0,1)$$ under $$T$$.

2. The linear transformation $$T:\mathbb R^2 \to \mathbb R^3$$ defined by $$T(x_1,x_2)=(x_1,x_2,0)$$ is one-to-one because $$T(x_1,x_2)=T(y_1,y_2) \implies (x_1,x_2,0)=(x_1,x_2,0) \implies (x_1,x_2)=(y_1,y_2)$$. But $$T$$ is not onto because $$(0,0,1)\in \mathbb R^3$$ has no pre-image $$(x_1,x_2)\in \mathbb R^2$$ under $$T$$.

3. The linear transformation $$T:\mathbb R^2 \to \mathbb R^2$$ defined by $$T(x_1,x_2)=(x_1+x_2,x_1-x_2)$$ is one-to-one and onto (exercise).

Theorem. Let $$T:\mathbb R^n \to \mathbb R^m$$ be a linear transformation with the standard matrix $$A$$. Then the following are equivalent.

1. $$T$$ (i.e., $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$) is one-to-one.

2. $$\ker T=\operatorname{NS}(A)=\{\overrightarrow{0_n}\}$$.

3. $$\operatorname{nullity}(T)=\operatorname{nullity}(A)=0$$.

4. The columns of $$A$$ are linearly independent.

(b), (c), and (d) are equivalent by the definitions.
(a) $$\implies$$ (b) Suppose $$T$$ (i.e., $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$) is one-to-one. Let $$\overrightarrow{x} \in \ker T=\operatorname{NS}(A)$$. Then $$A\overrightarrow{x}=\overrightarrow{0_m}$$. Also $$\overrightarrow{0_n} \mapsto A\overrightarrow{0_n}=\overrightarrow{0_m}$$. Since $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$ is one-to-one, $$\overrightarrow{x}=\overrightarrow{0_n}$$. Thus $\operatorname{NS}(A)=\{\overrightarrow{0_n}\}.$ (b) $$\implies$$ (a) Suppose $$\ker T=\operatorname{NS}(A)=\{\overrightarrow{0_n}\}$$. Let $$\overrightarrow{x},\overrightarrow{y} \in \mathbb R^n$$ such that $$A\overrightarrow{x}= A\overrightarrow{y}$$. Then $$A(\overrightarrow{x}-\overrightarrow{y})=\overrightarrow{0_m}$$. Then $$\overrightarrow{x}-\overrightarrow{y} \in \operatorname{NS}(A)=\{\overrightarrow{0_n}\}$$ which implies $$\overrightarrow{x}-\overrightarrow{y}=\overrightarrow{0_n}$$, i.e., $$\overrightarrow{x}=\overrightarrow{y}$$. Thus $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$ is one-to-one.

Example. The linear transformation $$T:\mathbb R^2 \to \mathbb R^3$$ defined by $$T(x_1,x_2)=(x_1,x_2,0)$$ has the standard matrix $$A=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) ] =\left[\begin{array}{rr} 1&0\\ 0&1\\ 0&0 \end{array} \right]$$. Note that the columns of $$A$$ are linearly independent, $$\ker T=\operatorname{NS}(A)=\{\overrightarrow{0_2}\}$$, and $$\operatorname{nullity}(T)=\operatorname{nullity}(A)=0$$. Thus $$T$$ (i.e., $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$) is one-to-one.

Theorem. Let $$T:\mathbb R^n \to \mathbb R^m$$ be a linear transformation with the standard matrix $$A$$. Then the following are equivalent.

1. $$T$$ (i.e., $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$) is onto.

2. $$\operatorname{im} T=\operatorname{CS}\left(A\right)=\mathbb R^m$$.

3. $$\operatorname{rank}(T)=\operatorname{rank}(A)=m$$.

4. Each row of $$A$$ has a pivot position.

(b), (c), and (d) are equivalent by the definitions.
(a) $$\implies$$ (b) Suppose $$T$$ (i.e., $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$) is onto. Let $$\overrightarrow{b} \in \mathbb R^m$$. Since $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$ is onto, $$\overrightarrow{b}=A\overrightarrow{x}$$ for some $$\overrightarrow{x}\in \mathbb R^n$$. Then $$\overrightarrow{b}=A\overrightarrow{x} \in \operatorname{CS}\left(A\right)$$. Thus $$\operatorname{im} T=\operatorname{CS}\left(A\right)=\mathbb R^m$$.
(b) $$\implies$$ (a) Suppose $$\operatorname{im} T=\operatorname{CS}\left(A\right)=\mathbb R^m$$. Let $$\overrightarrow{b} \in \mathbb R^m$$. Since $$\overrightarrow{b} \in \operatorname{CS}\left(A\right)=\mathbb R^m$$, $$\overrightarrow{b}=A\overrightarrow{x}$$ for some $$\overrightarrow{x}\in \mathbb R^n$$. Thus $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$ is onto.

Example. The linear transformation $$T:\mathbb R^3 \to \mathbb R^2$$ defined by $$T(x_1,x_2,x_3)=(x_1,x_2)$$ has the standard matrix $$A=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) \: T(\overrightarrow{e_3})] =\left[\begin{array}{rrr} 1&0&0\\ 0&1&0 \end{array} \right]$$. Note that each row of $$A$$ has a pivot position, $$\operatorname{im} T=\operatorname{CS}\left(A\right)=\mathbb R^2$$, and $$\operatorname{rank}(T)=\operatorname{rank}(A)=2$$. Thus $$T$$ (i.e., $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$) is onto.

Definition. A linear transformation $$T:\mathbb R^n \to \mathbb R^n$$ is an isomorphism if it is one-to-one and onto.

Example. The linear transformation $$T:\mathbb R^2 \to \mathbb R^2$$ defined by $$T(x_1,x_2)=(x_1+x_2,x_1-x_2)$$ is one-to-one and onto consequently an isomorphism. Showing $$T$$ is one-to-one is enough to show $$T$$ is an isomorphism by the following theorem.

Theorem. Let $$T:\mathbb R^n \to \mathbb R^n$$ be a linear transformation with the $$n\times n$$ standard matrix $$A$$. Then the following are equivalent.

1. $$T$$ (i.e., $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$) is an isomorphism.

2. $$T$$ (i.e., $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$) is one-to-one.

3. $$\ker T=\operatorname{NS}(A)=\{\overrightarrow{0_n}\}$$.

4. $$\operatorname{nullity}(T)=\operatorname{nullity}(A)=0$$.

5. The columns of $$A$$ are linearly independent.

6. $$T$$ (i.e., $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$) is onto.

7. $$\operatorname{im} T=\operatorname{CS}\left(A\right)=\mathbb R^n$$.

8. $$\operatorname{rank}(T)=\operatorname{rank}(A)=n$$.

9. Each row and column of $$A$$ has a pivot position.

(f), (g), (h), and (i) are equivalent by the preceding theorem. (b), (c), (d), and (e) are equivalent by the theorem before the preceding theorem. Now for the $$n\times n$$ standard matrix $$A$$, $$\operatorname{rank}(A)+\operatorname{nullity}(A)=n$$. Thus $$\operatorname{nullity}(A)=0$$ if and only if $$\operatorname{rank}(A)=n$$, i.e., (d) and (h) are equivalent. Since (b) and (f) are equivalent, they are equivalent to (a).

Example. What can we say about $$\operatorname{CS}\left(A\right),\operatorname{NS}(A),\operatorname{rank}(A), \operatorname{nullity}(A)$$, and pivot positions of a $$3\times 3$$ matrix with three linearly independent columns? What about $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$?

Solution. By the preceding theorem, $$\operatorname{CS}\left(A\right)=\mathbb R^3,\operatorname{NS}(A) =\{\overrightarrow{0_3}\},\operatorname{rank}(A)=3,\operatorname{nullity}(A)=0$$, $$A$$ has 3 pivot positions, and $$\overrightarrow{x} \mapsto A\overrightarrow{x}$$ is a one-to-one linear transformation from $$\mathbb R^3$$ onto $$\mathbb R^3$$.

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