Question

In Exercises 23 and 24, mark each statement True or False. Justify each answer.

24.

a. Not every linear transformation from\({\mathbb{R}^n}\)to\({\mathbb{R}^m}\)is a matrix transformation.

b. The columns of the standard matrix for a linear transformation from\({\mathbb{R}^n}\)to\({\mathbb{R}^m}\)are the images of the columns of the\(n \times n\)identity matrix.

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\(\left[ {\begin{array}{*{20}{c}}a&0\\0&d\end{array}} \right]\), where\(a\)and\(d\)are\( \pm 1\).

d. A mapping\(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\)is one-to-one if each vector in\({\mathbb{R}^n}\)maps onto a unique vector in\({\mathbb{R}^m}\).

e. If\(A\)is a\(3 \times 2\)matrix, then the transformation\(x \mapsto Ax\)cannot map\({\mathbb{R}^2}\)onto\({\mathbb{R}^3}\).

Short answer

a. The given statement is false.

b. The given statement is true.

c. The given statement is true.

d. The given statement is false.

e. The given statement is true.

Step-by-step solution

Determine whether the given statement is true or false

(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.

Determine whether the given statement is true or 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.

Determine whether the given statement is true or false

(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.

Determine whether the given statement is true or false

(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.

Determine whether the given statement is true or 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.