Question

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.

Short answer

a. The given statement is true.

b. The given statement is true.

c. The given statement is false.

d. The given statement is false.

e. The given statement is false.

Step-by-step solution

Determine whether the given statement is true or false

(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.

Determine whether the given statement is true or false

(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.

Determine whether the given statement is true or false

(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.

Determine whether the given statement is true or 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.

Determine whether the given statement is true or 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.