Question

In Exercise 22, mark each statement True or False. Justify each answer.

22. a. Every matrix transformation is a linear transformation.

b. The codomain of the transformation \({\bf{x}} \mapsto {\bf{Ax}}\) is the set of all linear combinations of the columns of \({\bf{A}}\).

c. If \({\bf{T}}:{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{m}}}\) is a linear transformation and if \({\bf{c}}\) is in \({\mathbb{R}^{\bf{m}}}\), then a uniqueness is “Is c in the range of T?”

d. A linear transformation preserves the operations of vector addition and scalar multiplication.

e. The superposition principle is a physical description of a linear transformation.

Short answer

1. The statement is true.
2. The statement is false.
3. The statement is false.
4. The statement is true.
5. The statement is true.

Step-by-step solution

Use the properties of matrices

(a)

Suppose \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) by \(T\left( x \right) = Ax\) is a matrix transformation, where \(A\) is a \(m \times n\) matrix.

Claim: \(T\) is linear.

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

\(\begin{array}{c}T\left( {u + v} \right) = Au + Av\\T\left( {u + v} \right) = T\left( u \right) + T\left( v \right)\end{array}\)

(ii) \(T\left( {cw} \right) = A\left( {cw} \right)\)

\(\begin{array}{c}T\left( {cw} \right) = cAw\\T\left( {cw} \right) = cT\left( w \right)\end{array}\)

From (i) and (ii), you have \(T\) is linear.

Hence, the given statement is true.

Use the definition of matrix transformation

(b)

Suppose \(A\) is a \(m \times n\) matrix. Then, the dimension of the codomain should be \(m\). However, the dimension of the set of all linear combinations of the columns of \(A\) need not be \(m\).

Hence, the given statement is false.

Use the definition of the range of \({\bf{T}}\)

(c)

The range of \(T\) is contained in \({\mathbb{R}^m}\). So, there always exists a \(c \in {\mathbb{R}^m}\) such that \(T\left( {ax + by} \right) \ne c\) for all scalars \(a,\,b\) and for all vectors \(x,\,y\) in \({\mathbb{R}^n}\). That is, \(c\) does not belong to the range of \(T\).

Hence, the given statement is false.

Use the definition of a linear transformation

(d)

Let \(T\) be linear. Then,

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

This implies \(T\) preserves the vector addition.

Also,

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

This implies \(T\) preserves the scalar multiplication.

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

Hence, the given statement is true.

Use the superposition rule

(e)

Let \(T\) be a linear transformation. Then,

\(T\left( {{c_1}{v_1} + {c_2}{v_2} + ... + {c_n}{v_n}} \right) = {c_1}T\left( {{v_1}} \right) + {c_2}T\left( {{v_2}} \right) + ... + {c_3}T\left( {{v_n}} \right)\)

It is known as the superposition rule. Think of \({v_1},{v_2},...,{v_n}\) as signals that go into the system and \(T\left( {{v_1}} \right),\,T\left( {{v_2}} \right),...,T\left( {{v_n}} \right)\) as the responses of that system to the signals.

Hence, the given statement is true.