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**1. The problem statement, all variables and given/known data**

I have to find the dimension of U and dim(V), of the union dim(U+V) and of dim(U[itex]\cap[/itex]V)

U is spanned by

\begin{align}

\begin{pmatrix}

1 \\

-2 \\

0

\end{pmatrix},

\begin{pmatrix}

1 \\

1 \\

2

\end{pmatrix}

\end{align} and V is spanned by

\begin{align}

\begin{pmatrix}

3 \\

0 \\

4

\end{pmatrix},

\begin{pmatrix}

0 \\

3 \\

a

\end{pmatrix}

\end{align} a[itex]\in[/itex][itex]\textbf{R}[/itex]

**2. Relevant equations**

[itex]dim(U)+dim(V)-dim(U [/itex][itex]\cap[/itex][itex]V)=dim(U+V) [/itex]

**3. The attempt at a solution**

Because the vectors spanning U and V are lin. independent:

[itex]dim(U) = dim(V) = 2 [/itex]

I find the intersection by equaling the two subspaces and then solving the linear system. But how do I find the sum of the two subspaces without calculating the intersection first?

Any hints are very appreciated :)