In Exercises 23 and 24, mark each statement True or False. \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\)Justify each answer.

23.

a. A linear transformation is completely determined by its effect on the columns of the \(n \times n\) identity matrix.

b. If \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\) rotates vectors about the origin through an angle \(\varphi \)then \(T\) is a linear transformation.

c. When two linear transformations are performed one after another, the combined effect may not always be a linear transformation.

d. A mapping \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) is onto \({\mathbb{R}^m}\) if every vector \(x\) in \({\mathbb{R}^n}\) maps onto some vector in \({\mathbb{R}^m}\).

e. If \(A\) is a \(3 \times 2\) matrix, then the transformation \(x \mapsto Ax\) cannot be one-to-one.

(a)

Theorem 10 states that let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation, then there exists a unique matrix \(A\) such that \(T\left( x \right) = Ax\) for all \(x\) in \({\mathbb{R}^n}\). \(A\) is the \(m \times n\) matrix whose \[j{\mathop{\rm th}\nolimits} \] column of the identity matrix in \({\mathbb{R}^n}\) : \(A = \left[ {\begin{array}{*{20}{c}}{T\left( {{e_1}} \right)}&{...}&{T\left( {{e_n}} \right)}\end{array}} \right]\).

Thus, the given statement (a) is true.

(b)

Example 3 states that let \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\) be the transformation that rotates each point in \({\mathbb{R}^2}\) about the origin through an angle \(\varphi \), then \(T\) is a linear transformation.

Thus, the given statement (b) is true.

(c)

Other transformations can be constructed by applying one transformation after another. Such a composition of a linear transformation is linear.

Thus, the given statement (c) is false.

(d)

A transformation \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) is said to be onto \({\mathbb{R}^m}\) if each \[{\mathop{\rm b}\nolimits} \] in \({\mathbb{R}^m}\) is the image of at least one \(x\) in \({\mathbb{R}^n}\).

The function \({\mathbb{R}^n}\) to \({\mathbb{R}^m}\), each vector is mapped onto another vector.

Thus, the given statement (d) is false.

(e)

\(A\)is a \(3 \times 2\) matrix. The columns of \(A\) span \({\mathbb{R}^3}\) if and only if \(A\) has 3 pivot columns. Since \(A\) has only 2 columns, the columns of \(A\) do not span \({\mathbb{R}^3}\) , and linear transformation are not onto \({\mathbb{R}^3}\).

Thus, the given statement (e) is false.