Definitions and Basic Properties

Throughout this chapter, the scalar field ${\mathbb{F}}$ is either always the set ${\mathbb{R}}$ or always the set ${\mathbb{C}}.$

DEFINITION 4.1.1 (Linear Transformation)   Let $V$ and $W$ be vector spaces over ${\mathbb{F}}.$ A map $T: V {\longrightarrow}W$ is called a linear transformation if

$$T(\alpha {\mathbf u}+ \beta {\mathbf v}) = \alpha T({\mathbf u}) ... ...ha, \beta \in {\mathbb {F}}, \; \; {\mbox{and }} {\mathbf u},{\mathbf v}\in V.$$

We now give a few examples of linear transformations.

EXAMPLE 4.1.2  

1. Define $T: {\mathbb{R}}{\longrightarrow}{\mathbb{R}}^2$ by $\; T(x) = (x, 3x)$ for all $x \in {\mathbb{R}}.$ Then $T$ is a linear transformation as
   $$T(x + y) = (x+y, 3(x+y)) = (x,3x) + (y,3y) = T(x) + T(y).$$

2. Verify that the maps given below from ${\mathbb{R}}^n$ to ${\mathbb{R}}$ are linear transformations. Let ${\mathbf x}= (x_1, x_2, \ldots, x_n).$
   1. Define $\; T( {\mathbf x}) = \sum\limits_{i=1}^n x_i.$
   2. For any $i, \; \; 1 \leq i \leq n,$ define $\; T_i( {\mathbf x}) = x_i.$
   3. For a fixed vector ${\mathbf a}= (a_1, a_2, \ldots, a_n) \in {\mathbb{R}}^n,$ define $\; T( {\mathbf x}) = \sum\limits_{i=1}^n a_i x_i.$ Note that examples $(a)$ and $(b)$ can be obtained by assigning particular values for the vector ${\mathbf a}.$
3. Define $T: {\mathbb{R}}^2 {\longrightarrow}{\mathbb{R}}^3$ by $\; T((x,y)) = (x+y, 2 x - y, x + 3 y).$
   Then $T$ is a linear transformation with $T((1,0)) = (1,2,1)$ and $T((0,1)) = (1,-1,3).$
4. Let $A$ be an $m \times n$ real matrix. Define a map $T_A: {\mathbb{R}}^n {\longrightarrow}{\mathbb{R}}^m$ by
   $$T_A({\mathbf x}) = A {\mathbf x}\;\; {\mbox{ for every }} \;\; {\mathbf x}^{t} = (x_1, x_2, \ldots, x_n) \in {\mathbb{R}}^n.$$
   Then $T_A$ is a linear transformation. That is, every $m \times n$ real matrix defines a linear transformation from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m.$
5. Recall that ${\cal P}_n({\mathbb{R}})$ is the set of all polynomials of degree less than or equal to $n$ with real coefficients. Define $\; T: {\mathbb{R}}^{n+1} {\longrightarrow} {\cal P}_n ({\mathbb{R}})$ by
   $$T( (a_1, a_2, \ldots, a_{n+1}) ) = a_1 + a_2 x + \cdots + a_{n+1} x^n$$
   for $\; (a_1, a_2, \ldots, a_{n+1}) \in {\mathbb{R}}^{n+1}.$ Then $T$ is a linear transformation.

PROPOSITION 4.1.3   Let $T: V {\longrightarrow}W$ be a linear transformation. Suppose that ${\mathbf 0}_V$ is the zero vector in $V$ and ${\mathbf 0}_W$ is the zero vector of $W.$ Then $T({\mathbf 0}_V) = {\mathbf 0}_W.$

Proof. Since ${\mathbf 0}_V = {\mathbf 0}_V + {\mathbf 0}_V,$ we have

$$T({\mathbf 0}_V) = T({\mathbf 0}_V + {\mathbf 0}_V)= T({\mathbf 0}_V) + T({\mathbf 0}_V).$$

So, $T({\mathbf 0}_V) = {\mathbf 0}_W$ as $T({\mathbf 0}_V) \in W.$ height6pt width 6pt depth 0pt

From now on, we write ${\mathbf 0}$ for both the zero vector of the domain space and the zero vector of the range space.

DEFINITION 4.1.4 (Zero Transformation)   Let $V$ be a vector space and let $T: V {\longrightarrow}W$ be the map defined by

$$T({\mathbf v}) = {\mathbf 0}\; {\mbox{ for every }} \; {\mathbf v}\in V.$$

Then $T$ is a linear transformation. Such a linear transformation is called the zero transformation and is denoted by ${\mathbf 0}.$

DEFINITION 4.1.5 (Identity Transformation)   Let $V$ be a vector space and let $T: V {\longrightarrow}V$ be the map defined by

$$T({\mathbf v}) = {\mathbf v}\; {\mbox{ for every }} \; {\mathbf v}\in V.$$

Then $T$ is a linear transformation. Such a linear transformation is called the Identity transformation and is denoted by $I.$

We now prove a result that relates a linear transformation $T$ with its value on a basis of the domain space.

THEOREM 4.1.6   Let $T: V {\longrightarrow}W$ be a linear transformation and ${\cal B}= ( {\mathbf u}_1, {\mathbf u}_2, \ldots, {\mathbf u}_n )$ be an ordered basis of $V.$ Then the linear transformation $T$ is a linear combination of the vectors $T({\mathbf u}_1), T({\mathbf u}_2), \ldots, T({\mathbf u}_n).$

In other words, $T$ is determined by $T({\mathbf u}_1), T({\mathbf u}_2), \ldots, T({\mathbf u}_n).$

Proof. Since ${\cal B}$ is a basis of $V,$ for any ${\mathbf x}\in V,$ there exist scalars $\alpha_1, {\alpha}_2, \ldots, \alpha_n$ such that ${\mathbf x}= \alpha_1 {\mathbf u}_1 + \alpha_2 {\mathbf u}_2 + \cdots + \alpha_n {\mathbf u}_n.$ So, by the definition of a linear transformation

$$T({\mathbf x}) = T(\alpha_1 {\mathbf u}_1 + \cdots + \alpha_n {\mathbf u}_n) = \alpha_1 T({\mathbf u}_1) + \cdots + \alpha_n T({\mathbf u}_n).$$

Observe that, given ${\mathbf x}\in V,$ we know the scalars ${\alpha}_1, {\alpha}_2, \ldots, {\alpha}_n.$ Therefore, to know $T({\mathbf x}),$ we just need to know the vectors $T({\mathbf u}_1), T({\mathbf u}_2), \ldots, T({\mathbf u}_n)$ in $W.$

That is, for every ${\mathbf x}\in V,$ $T({\mathbf x})$ is determined by the coordinates $(\alpha_1, \alpha_2, \ldots, \alpha_n)$ of ${\mathbf x}$ with respect to the ordered basis ${\cal B}$ and the vectors $T({\mathbf u}_1), T({\mathbf u}_2), \ldots, T({\mathbf u}_n) \in W.$ height6pt width 6pt depth 0pt

EXERCISE 4.1.7  

1. Which of the following are linear transformations $T : V {\longrightarrow}W?$ Justify your answers.
   1. Let $V = {\mathbb{R}}^2$ and $W = {\mathbb{R}}^3$ with $T\bigl( \; (x,y) \; \bigr) = (x + y + 1, 2 x - y, x + 3 y)$
   2. Let $V = W = {\mathbb{R}}^2$ with $T \bigl( \;(x,y) \; \bigr) = (x-y, x^2 - y^2)$
   3. Let $V = W = {\mathbb{R}}^2$ with $T \bigl( \;(x,y) \; \bigr) = (x-y, \vert x\vert)$
   4. Let $V = {\mathbb{R}}^2$ and $W = {\longrightarrow}{\mathbb{R}}^4$ with $T\bigl( \;(x,y) \; \bigr)= (x + y, x-y, 2x + y, 3x - 4 y)$
   5. Let $V = W = {\mathbb{R}}^4$ with $T\bigl( \;(x,y,z,w)\; \bigr)= (z,x,w,y)$
2. Recall that $M_2({\mathbb{R}})$ is the space of all $2 \times 2$ matrices with real entries. Then, which of the following are linear transformations $T : M_2({\mathbb{R}}) {\longrightarrow}M_2({\mathbb{R}})?$

   $(a) \;\; T(A) = A^t \hspace{1in} (b) \;\; T(A) = I + A \hspace{1in} (c) \;\; T(A) = A^2$

   $(d) \;\; T(A) = B A B^{-1},$ where $B$ is some fixed $2 \times 2$ matrix.

3. Let $T: {\mathbb{R}} \longrightarrow {\mathbb{R}}$ be a map. Then $T$ is a linear transformation if and only if there exists a unique $c \in {\mathbb{R}}$ such that $\; T({\mathbf x}) = c {\mathbf x}$ for every ${\mathbf x}\in {\mathbb{R}}.$
4. Let $A$ be an $n \times n$ real matrix. Consider the linear transformation
   $$T_A({\mathbf x}) = A {\mathbf x}\;\; {\mbox{ for every }} \; {\mathbf x}\in {\mathbb{R}}^n.$$
   Then prove that $T^2({\mathbf x}) := T(T({\mathbf x})) = A^2 {\mathbf x}.$ In general, for $k \in {\mathbb{N}},$ prove that $T^k ({\mathbf x}) = A^k {\mathbf x}.$
5. Use the ideas of matrices to give examples of linear transformations $T,S : {\mathbb{R}}^3 {\longrightarrow}{\mathbb{R}}^3$ that satisfy:
   1. $T \neq {\mathbf 0}, \;\; T^2 \neq {\mathbf 0}, \;\; T^3 = {\mathbf 0}.$
   2. $T \neq {\mathbf 0}, \;\; S \neq {\mathbf 0}, \;\; S\circ T \neq {\mathbf 0}, \;\; T \circ S = {\mathbf 0};$ where $T\circ S({\mathbf x}) = T\bigl(S({\mathbf x})\bigr).$
   3. $S^2 = T^2, \;\; S \neq T.$
   4. $T^2 = I, \;\; T \neq I.$
6. Let $T : {\mathbb{R}}^n \longrightarrow {\mathbb{R}}^n$ be a linear transformation such that $T \neq {\mathbf 0}$ and $T^2 = {\mathbf 0}.$ Let ${\mathbf x}\in {\mathbb{R}}^n$ such that $T ({\mathbf x}) \neq {\mathbf 0}.$ Then prove that the set $\{{\mathbf x}, T({\mathbf x})\}$ is linearly independent. In general, if $T^k \neq {\mathbf 0}$ for $1 \leq k \leq p$ and $T^{p+1} = {\mathbf 0},$ then for any vector ${\mathbf x}\in {\mathbb{R}}^n$ with $T^p ({\mathbf x}) \neq {\mathbf 0}$ prove that the set $\{{\mathbf x}, T({\mathbf x}), \ldots, T^p ({\mathbf x}) \}$ is linearly independent.
7. Let $T : {\mathbb{R}}^n \longrightarrow {\mathbb{R}}^m$ be a linear transformation, and let ${\mathbf x}_0 \in {\mathbb{R}}^n$ with $T({\mathbf x}_0) = y.$ Consider the sets
   $$S = \{ {\mathbf x}\in {\mathbb{R}}^n : T({\mathbf x}) = {\mathbf ... ... }} \;\; N = \{ {\mathbf x}\in {\mathbb{R}}^n : T({\mathbf x}) = {\mathbf 0}\}.$$
   Show that for every ${\mathbf x}\in S$ there exists ${\mathbf z}\in N$ such that ${\mathbf x}= {\mathbf x}_0 + z.$
8. Define a map $T: {\mathbb{C}} \longrightarrow {\mathbb{C}}$ by $T(z) = \overline{z},$ the complex conjugate of $z.$ Is $T$ linear on

   (a) $\;$ ${\mathbb{C}}$ over ${\cal R}\;\;$ (b) $\;$ ${\mathbb{C}}$ over ${\mathbb{C}}.$

9. Find all functions $f : {\mathbb{R}}^2 \longrightarrow {\mathbb{R}}^2$ that satisfy the conditions
   1. $f(\;(x,x)\;) = (x,x)$ and
   2. $f(\;(x,y)\;) = (y,x)$ for all $(x,y) \in {\mathbb{R}}^2.$
   That is, $f$ fixes the line $y = x$ and sends the point $(x_1, y_1)$ for $x_1 \neq y_1$ to its mirror image along the line $y = x.$

   Is this function a linear transformation? Justify your answer.

THEOREM 4.1.8   Let $T: V {\longrightarrow}W$ be a linear transformation. For ${\mathbf w}\in W,$ define the set

$$T^{-1}({\mathbf w}) = \{{\mathbf v}\in V : T({\mathbf v}) = {\mathbf w}\}.$$

Suppose that the map $T$ is one-one and onto.

1. Then for each ${\mathbf w}\in W,$ the set $T^{-1}({\mathbf w})$ is a set consisting of a single element.
2. The map $T^{-1} : W {\longrightarrow}V$ defined by
   $$T^{-1}({\mathbf w}) = {\mathbf v}\; {\mbox{ whenever }} \; T({\mathbf v}) = {\mathbf w}.$$
   is a linear transformation.

Proof. Since $T$ is onto, for each ${\mathbf w}\in W$ there exists a vector ${\mathbf v}\in V$ such that $T({\mathbf v}) = {\mathbf w}.$ So, the set $T^{-1}({\mathbf w})$ is non-empty.

Suppose there exist vectors ${\mathbf v}_1, {\mathbf v}_2 \in V$ such that $T({\mathbf v}_1) = T({\mathbf v}_2).$ But by assumption, $T$ is one-one and therefore ${\mathbf v}_1 = {\mathbf v}_2.$ This completes the proof of Part $1.$

We now show that $T^{-1}$ as defined above is a linear transformation. Let ${\mathbf w}_1, {\mathbf w}_2 \in W.$ Then by Part $1,$ there exist unique vectors ${\mathbf v}_1, {\mathbf v}_2 \in V$ such that $T^{-1}({\mathbf w}_1) = {\mathbf v}_1$ and $T^{-1}({\mathbf w}_2) = {\mathbf v}_2.$ Or equivalently, $T({\mathbf v}_1) = {\mathbf w}_1$ and $T({\mathbf v}_2) = {\mathbf w}_2.$ So, for any ${\alpha}_1, {\alpha}_2 \in {\mathbb{F}},$ we have $T({\alpha}_1{\mathbf v}_1 + {\alpha}_2{\mathbf v}_2) = {\alpha}_1 {\mathbf w}_1 + {\alpha}_2 {\mathbf w}_2.$

Thus for any ${\alpha}_1, {\alpha}_2 \in {\mathbb{F}},$

$$T^{-1}({\alpha}_1{\mathbf w}_1 + {\alpha}_2{\mathbf w}_2) = {\alp... ...bf v}_2 = {\alpha}_1 T^{-1}({\mathbf w}_1) + {\alpha}_2 T^{-1}({\mathbf w}_2).$$

Hence $T^{-1}:W {\longrightarrow}V,$ defined as above, is a linear transformation. height6pt width 6pt depth 0pt

DEFINITION 4.1.9 (Inverse Linear Transformation)   Let $T: V {\longrightarrow}W$ be a linear transformation. If the map $T$ is one-one and onto, then the map $T^{-1} : W {\longrightarrow}V$ defined by

$$T^{-1}({\mathbf w}) = {\mathbf v}\;\; {\mbox{ whenever }} T({\mathbf v}) = {\mathbf w}$$

is called the inverse of the linear transformation $T.$

EXAMPLE 4.1.10  

1. Define $T: {\mathbb{R}}^2 {\longrightarrow}{\mathbb{R}}^2$ by $T((x,y)) = (x+y, x-y).$ Then $T^{-1}: {\mathbb{R}}^2 {\longrightarrow}{\mathbb{R}}^2$ is defined by
   $$T^{-1}((x,y)) = (\displaystyle\frac{x+y}{2}, \displaystyle\frac{x-y}{2}).$$
   Note that
   

   $$T\circ T^{-1} ((x,y)) = T(T^{-1}((x,y))) = T( (\frac{x+y}{2}, \frac{x-y}{2}) )$$

   $$= (\frac{x+y}{2}+ \frac{x-y}{2}, \frac{x+y}{2}- \frac{x-y}{2})$$

   $$= (x,y).$$

   
   Hence, $T\circ T^{-1} = I,$ the identity transformation. Verify that $T^{-1} \circ T = I.$ Thus, the map $T^{-1}$ is indeed the inverse of the linear transformation $T.$
2. Recall the vector space ${\cal P}_n({\mathbb{R}})$ and the linear transformation $\; T: {\mathbb{R}}^{n+1} {\longrightarrow} {\cal P}_n ({\mathbb{R}})$ defined by
   $$T( (a_1, a_2, \ldots, a_{n+1}) ) = a_1 + a_2 x + \cdots + a_{n+1} x^n$$
   for $(a_1, a_2, \ldots, a_{n+1}) \in {\mathbb{R}}^{n+1}.$ Then $\; T^{-1}: {\cal P}_n({\mathbb{R}}) {\longrightarrow}{\mathbb{R}}^{n+1}$ is defined as
   $$T^{-1}( a_1 + a_2 x + \cdots + a_{n+1} x^n) = (a_1, a_2, \ldots, a_{n+1})$$
   for $a_1+ a_2 x + \cdots + a_{n+1} x^n \in {\cal P}_n({\mathbb{R}}).$ Verify that $T\circ T^{-1} = T^{-1} \circ T = I.$ Hence, conclude that the map $T^{-1}$ is indeed the inverse of the linear transformation $T.$