Given the following matrices

Ap$\backslash$times q $=[$ a$1$ a$2$ aq $],$ Bq$\backslash$times r $=$

$$

$$

$$

$$

$$

$$

b

T

$1$

b

T

$2$

$.$

$.$

$.$

b

T

q

$$

$$

$$

$$

$$

$$

$,$ D $=$ diag$\{$d$1,$ d$2,,$ dq$\},$

where D is a diagonal matrix with di on the main diagonal and zeros elsewhere. Use

the facts of matrix multiplication to show that

A $$ D $$ B $=$

X

q

i$=1$

diaib

T

i

$.$

n

$.$ Let V $=\{$x$1\}$ be a set containing one

nonzero vector from S$.$ Is V a basis for S$?$ Explain your answer