I need help with determining the basis of $U_1 \cap U_2$ in the following problem:
Let $V=\mathbb{R}^4$. ${U_1} = \left\{ {\left( {\begin{array}{*{20}{c}}
{{x_1}} \\
{{x_2}} \\
{{x_3}} \\
{{x_4}}
\end{array}} \right)\left| {{x_1} – {x_2} + {x_3} – 3{x_4} = 0} \right.} \right\}$ and $U_2=\left\langle {\left( {\begin{array}{*{20}{c}}
1 \\
1 \\
0 \\
3
\end{array}} \right),\left( {\begin{array}{*{20}{c}}
0 \\
{ – 1} \\
0 \\
1
\end{array}} \right)} \right\rangle$.
If $U_1$ is a subspace of $V$, determine a basis of $U_1 \cap U_2$.
My attempt:
I know that ${U_2} = \left\{ {\left( {\begin{array}{*{20}{c}}
\lambda \\
{\lambda – \mu } \\
0 \\
{3\lambda + \mu }
\end{array}} \right)\left| {\lambda ,\mu \in \mathbb{R}} \right.} \right\}$, and that the next step is that I should choose an element in $U_1$ and in $U_2$, e.g. Let $w \in {U_1}$ and let $w \in {U_2}$. Then we know that $w$ is of the form $w = \left( {\begin{array}{*{20}{c}}
\lambda \\
{\lambda – \mu } \\
0 \\
{3\lambda + \mu }
\end{array}} \right)$, but I'm not sure what the procedure is from there.
Best Answer
The next step is to note that\begin{align}U_1\cap U_2&=\left\{\begin{pmatrix}\lambda\\\lambda-\mu\\0\\3\lambda+\mu\end{pmatrix}\,\middle|\,\lambda-(\lambda-\mu)-3(3\lambda+\mu)=0\right\}\\&=\left\{\begin{pmatrix}\lambda\\\lambda-\mu\\0\\3\lambda+\mu\end{pmatrix}\,\middle|\,9\lambda+2\mu=0\right\}\\&=\left\{\begin{pmatrix}\lambda\\\frac{11}2\lambda\\0\\-\frac32\lambda\end{pmatrix}\,\middle|\,\lambda\in\mathbb{R}\right\}.\end{align}Can you take it from here?