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Linear Algebra

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Moderate

Dimension of Direct Sum

LINALG-JGKEWG

Suppose $V$ and $W$ are subspaces of $\mathbb{R}^n$ where $V\cap W=\{\vec{0}\}$ and $\{\vec{v}_1,\dots,\vec{v}_p\}$ is a basis for $V$ and $\{\vec{w}_1,\dots,\vec{w}_q\}$ is a basis for $W$.

If:

$$Z=Span(\{\vec{v}_1,\dots,\vec{v}_p,\vec{w}_1,\dots,\vec{w}_q\})$$

...then:

A

$\dim(Z) < p+q$

B

$\dim(Z)>p+q$

C

$\dim(Z)=p+q$

D

$\dim(Z)$ cannot be determined without more information.