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Linear Algebra

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Moderate

Coordinate Vector

LINALG-JZDLEE

Suppose that:

$$S=\{\vec v_1, \vec v_2, \ldots, \vec v_n\}$$

...is a basis for $V$. Suppose $[\vec w]_S$ denotes the coordinate vector of $\vec w$ with respect to $S$.

Which of the following statements must be true? Select ALL that apply.

A

For all $\vec w$ in $V$, $[\vec w]_S$ exists.

B

For all non-zero $\vec w$ in $V$, $[\vec w]_S$ exists.

C

If $[\vec w]_S$ exists, it is unique.

D

If $[\vec w]_S=\vec 0$, then $\vec w=\vec 0$.

E

It is possible that $\vec v$ and $\vec w$ are unequal elements of $V$ and $[\vec v]_S=[\vec w]_S$.

F

None of the above.