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Linear Algebra

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Moderate

New Bases from Old

LINALG-X4BEKJ

If $B=\{\vec{v}_1,\vec{v}_2,\vec{v}_3\}$ is a basis for a vector space $V$, then let: $S=\{\vec{w}_1,\vec{w}_2,\vec{w}_3\}$

...where:

$\vec{w}_1=\vec{v}_1$
$\vec{w}_2=\vec{v}_1+\vec{v}_2$
$\vec{w}_3=\vec{v}_1+\vec{v}_2+\vec{v}_3$

This set $S$:

A

Is linearly independent, but does not span $V$

B

Spans $V$ but is not linearly independent

C

Is a basis for $V$

D

May or may not be a basis for $V$, it can't be determined