Question 4.1.2: Let A be an m × n matrix. Define a mapping T: R^n → R^m by T...
Let A be an m × n matrix. Define a mapping T: R^{n} → R^{m} by
T (x) = Ax
a. Show that T is a linear transformation.
b. Let A be the 2 ×3 matrix
A= \begin{bmatrix} 1& 2&-1 \\ -1&3&2 \end{bmatrix}
Find the images of
\begin{bmatrix} 1 \\1 \\1 \end{bmatrix} and \begin{bmatrix} 7 \\-1 \\5 \end{bmatrix}under the mapping T: R³ → R² with T (x) = Ax.
a. By Theorem 5 of Sec. 1.3, for all vectors u and v in R^{n} and all scalars c in R,
A(cu + v) = cAu + Av
Therefore,
T (cu + v) = cT (u) + T (v)
b. Since T is defined by matrix multiplication, we have
T\left(\begin{bmatrix} 1 \\1 \\1 \end{bmatrix} \right) = \begin{bmatrix} 1& 2&-1 \\ -1&3&2 \end{bmatrix} \begin{bmatrix} 1 \\1 \\1 \end{bmatrix} = \begin{bmatrix} 2 \\4 \end{bmatrix}
and
T\left(\begin{bmatrix} 7 \\-1 \\5 \end{bmatrix} \right) = \begin{bmatrix} 1& 2&-1 \\ -1&3&2 \end{bmatrix} \begin{bmatrix} 7\\-1 \\5 \end{bmatrix} = \begin{bmatrix} 0\\0 \end{bmatrix}