In Exercise 21, mark each statement True or False. Justify each answer.\(\) \(\)

21. a. A linear transformation is a special type of function.

b. If \({\bf{A}}\) is a \({\bf{3}} \times {\bf{5}}\)matrix and \({\bf{T}}\) is a transformation defined by\({\bf{T}}\left( {\bf{x}} \right) = {\bf{Ax}}\), then the domain of \({\bf{T}}\) is\({\mathbb{R}^{\bf{3}}}\).

c. If \({\bf{A}}\) is an \({\bf{m}} \times {\bf{n}}\) matrix, then the range of the transformation \({\bf{x}} \mapsto {\bf{Ax}}\) is\({\mathbb{R}^{\bf{m}}}\).

\(\) d. Every linear transformation is a matrix transformation.

e. A transformation \({\bf{T}}\) is linear if and only if \({\bf{T}}\left( {{{\bf{c}}_{\bf{1}}}{{\bf{v}}_{\bf{1}}} + {{\bf{c}}_{\bf{2}}}{{\bf{v}}_{\bf{2}}}} \right) = {{\bf{c}}_{\bf{1}}}{\bf{T}}\left( {{{\bf{v}}_{\bf{1}}}} \right) + {{\bf{c}}_{\bf{2}}}{\bf{T}}\left( {{{\bf{v}}_{\bf{2}}}} \right)\) for all \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) in the domain of \({\bf{T}}\) and for all scalars \({{\bf{c}}_{\bf{1}}}\) and\({{\bf{c}}_{\bf{2}}}\).\[\]

(a)

Note that every linear transformation is a function. However, not every function is a linear transformation. When a function satisfies the following axioms, only then will it be linear.

i.e., (i) \(T\left( {u + v} \right) = T\left( u \right) + T\left( v \right)\)

(ii) \(T\left( {cw} \right) = cT\left( w \right)\)

Here, c is a scalar and u, v, and w are the vectors in the domain of T.

The above axioms make the linear transformation a special function.

Hence, the given statement is true.

(b)

Given, \(A\) is a \(3 \times 5\) matrix. This implies \(A\) contains 5 column vectors. Since \(T\left( x \right) = Ax\), the domain of \(T\)is \({\mathbb{R}^5}\).

Thus, the given statement is false.

(c)

The given matrix transformation is \(x \mapsto Ax\) and \(A\) is a \(m \times n\) matrix. This implies its domain is \({\mathbb{R}^n}\) and its codomain is \({\mathbb{R}^m}\). Note that the range of the transformation is the set of all linear combinations of the column vectors in \(A\). This implies the range is contained in \({\mathbb{R}^m}\) but not exactly\({\mathbb{R}^m}\). This is possible if and only if the transformation is onto.

Hence, the given statement is false.

(d)

Every matrix transformation is a linear transformation. However, the converse need not be true. Hence, every linear transformation is not a matrix transformation.

Thus, the given statement is false.

(e)

Suppose \(T\) is a linear transformation. This implies

. (i) \(T\left( {u + v} \right) = T\left( u \right) + T\left( v \right)\)

(ii) \(T\left( {cw} \right) = cT\left( w \right)\)

Consider,

\(\begin{aligned}{c}T\left( {{c_1}{v_1} + {c_2}{v_2}} \right) &= T\left( {{c_1}{v_1}} \right) + T\left( {{c_2}{v_2}} \right)\\ &= {c_1}T\left( {{v_1}} \right) + {c_2}T\left( {{v_2}} \right)\end{aligned}\)

Suppose \(T\left( {{c_1}{v_1} + {c_2}{v_2}} \right) = {c_1}T\left( {{v_1}} \right) + {c_2}T\left( {{v_2}} \right)\)

This implies \(T\left( {{c_1}{v_1} + {c_2}{v_2}} \right) = T\left( {{c_1}{v_1}} \right) + T\left( {{c_2}{v_2}} \right)\).

Thus, \(T\) is linear.

Hence, the given statement is true.