?

Linear Algebra

Free Version

Upgrade subject to access all content

Moderate

Another Direct Sum Example

LINALG-NBEKEB

If:

$$V=\left\{a\begin{pmatrix} 1 \\\ 1\\\ 0\end{pmatrix}\ |\ a\in\mathbb{R}\right\}$$

...and:

$$W=\left\{b\begin{pmatrix} 0 \\\ 1\\\ 0\end{pmatrix}+c\begin{pmatrix} 0 \\\ 0\\\ 1\end{pmatrix}\ |\ b,c\in\mathbb{R}\right\}$$

...and if $Z=V+W$ where:

$$V+W=\{\vec{v}+\vec{w}\ |\ \vec{v}\in V\text{ and }\vec{w}\in W\}$$

...then:

A

$\dim(V\cap W)=1$ and $\dim(Z)=2$

B

$\dim(V\cap W)=0$ and $\dim(Z)=3$

C

$\dim(V\cap W)=2$ and $\dim(Z)=2$

D

$\dim(V\cap W)=3$ and $\dim(Z)=0$