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## Homework Statement

[PLAIN]http://img571.imageshack.us/img571/1821/subspaces.png [Broken]

## Homework Equations

## The Attempt at a Solution

Is my solution correct?:

For [itex]a,b\in \mathbb{C}[/itex]

let [itex]A=\begin{bmatrix} a \\ a \\ 0 \end{bmatrix}\in U[/itex] and [itex]B=\begin{bmatrix} 0 \\ b \\ b \end{bmatrix}\in W[/itex]

Then [itex]A+B=\begin{bmatrix} a \\ a \\ 0 \end{bmatrix} + \begin{bmatrix} 0 \\ b \\ b \end{bmatrix} = \begin{bmatrix} a \\ a+b \\ b \end{bmatrix}\in \mathbb{C}^3[/itex]

How do I get from this that [itex]U+W=V[/itex] ?

Clearly the only vector in the intersection of U and W is the zero vector when [itex]a=b=0[/itex] so [itex]U\cap W = \{\bf 0} \}[/itex]

[itex]v = \begin{bmatrix} a \\ b \\ c \end{bmatrix} \in A \cap B \Rightarrow \begin{cases} a=b, c=0 \quad v \in A \\ a = 0, b=c \quad v \in B \end{cases} \Rightarrow a=b=c=0[/itex]

[itex]\therefore V = U\oplus W[/itex]

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