Question

Compute the products Ax in Exercises 13 through 15 using

paper and pencil. In each case, compute the product

two ways: in terms of the columns of A and in terms of the rows of A.

15.

$\left[1234\right]\left[\begin{array}{c}5\\ 6\\ 7\\ 8\end{array}\right]$

Short answer

The dot product $A{x}^{\to }$of $[1234]\left[\begin{array}{c}5\\ 6\\ 7\\ 8\end{array}\right]$is [70].

Step-by-step solution

Step 1:

Find $A\overline{x}.$in terms of rows of A.

$[1234]\left[\begin{array}{c}5\\ 6\\ 7\\ 8\end{array}\right]$

$$\begin{gathered} \Rightarrow \left[\left(1\right)\left(5\right)+\left(2\right)\left(6\right)+\left(3\right)\left(7\right)+\left(4\right)\left(8\right)\right] \\ \Rightarrow \left[5+12+21+32\right] \\ \Rightarrow \left[70\right] \end{gathered}$$

Step 2:

Find $A\overline{x}.$ in terms of columns of A.

$[1234]\left[\begin{array}{c}5\\ 6\\ 7\\ 8\end{array}\right]$

$$\begin{gathered} \Rightarrow 5\left[1\right]+6\left[2\right]+7\left[3\right]+8\left[4\right] \\ \Rightarrow \left[5+12+21+32\right] \\ \left[70\right] \end{gathered}$$