Linear Algebra

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Moderate

Span of a Set of 2 $\times $ 3 Matrices

LINALG-ZEWGO7

Consider the set $S$, which is a subset of the set of all $2\times 3 $ matrices with real entries:

$$S=\bigg \{ \left [\matrix{ 1 & 0 & 1\cr 0 & 0 & 1}\right], \left [\matrix { 1 & 1& 1\cr 1 & 0 & 1}\right ], \left [\matrix { 1 & 1 & 0\cr 1 & 1 & 0}\right ]\bigg \}$$

Which of the following statements are correct? Choose ALL that apply.

A

$\left [\matrix { 0 & 0 & 0\cr 0 & 0 & 0}\right ]$

...is not in the span of $S$.

B

$\left [\matrix { 2 & 1 & 3\cr 1 & 2 & 1}\right ]$

...is in the span of $S$.

C

$\left [\matrix {1 & 0 & 0 \cr 0 & 1 & 0}\right ]$

...is in the span of $S$.

D

$\left [\matrix {6 & 5 & 4 \cr 5 & 3 & 4}\right ]$

...is in the span of $S$.

E

$\left [\matrix {-1 & 1 & 1 \cr 1 & -2 &1}\right ]$

...is in the span of $S$.

F

$\left [\matrix{4 & 1 & 2 \cr 2 & 3 & 2}\right ]$

...is not in the span of $S$.