91Ó°ÊÓ

(a)

Every linear transformation from\({\mathbb{R}^n}\)to\({\mathbb{R}^m}\)can be viewed as a matrix transformation.

Thus, the given statement (a) is false.

(b)

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 (b) is true.

(c)

The standard matrix of a linear transformation from\({\mathbb{R}^2}\)to\({\mathbb{R}^2}\)that reflects points through the horizontal axis, the vertical axis or the origin has the form\(\left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right]\).

Thus, the given statement (c) is true.

(d)

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

Thus, the given statement (d) is false.

(e)

Consider that\(A\)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 two columns, so the columns of\(A\)do not span\({\mathbb{R}^3}\), and the linear transformation does not map\({\mathbb{R}^2}\)onto\({\mathbb{R}^3}\).

Thus, the given statement (e) is true.