Hey I have the following exercise where I am having a problem. Can someone help me?
Consider the following vectors in the$ \mathbb{R}$ vector space $\mathbb{R}^3$: $v_1 =\begin{pmatrix}1\\1\\2\end{pmatrix}, v_2 =\begin{pmatrix}0\\1\\1\end{pmatrix},v_3 =\begin{pmatrix}2\\-1\\1\end{pmatrix},v_4 =\begin{pmatrix}1\\2\\1\end{pmatrix}$
Let $U = <v_1, v_2>$ and $V = <v_3, v_4>$.
(a) Determine a basis of $U$, of $V$ , of $U \cap V$ and of $U + V$ .
b)Give a complement of $U \cap V$ in $U + V$.
a) So here I found the basis of $U \cap V=<\begin{pmatrix}2\\-1\\1\end{pmatrix}>$ and for $U+V=<\begin{pmatrix}1\\1\\2\end{pmatrix},\begin{pmatrix}0\\1\\1\end{pmatrix},\begin{pmatrix}0\\0\\2\end{pmatrix}>$ (If you want you can check the results but they should be right)
For b) I really don't know how to proceed. What is asked is to find two vectors that are not in the basis of $U \cap V$ that added to the basis of $U \cap V$ gives a basis of $U+V$, right?
How can I do it?
Any help is welcome. Thanks!