I am having troubles trying to solve this example:
In $\mathbb{R}^4$, find a basis of $L1 \cap L2$, when $L_1=\operatorname{Span}\{a, b, c\}$ and $L_2=\operatorname{Span}\{d,e, f\}$ where:
- $a = (1, 1, 0, 2)$
- $b = (-1, 1, 2, 3)$
- $c = (-2, 0, 2, 1)$
- $d = (2, -1, 3, 1)$
- $e = (0, 1, 1, 2)$
- $f = (2, 2, 2, 1)$.
I tried solving this as an equation $αa + βb + γc = δd + εe + ζf$, but it does not look like the way to solve the problem.
The solution is $\operatorname{Span}\{-1, 5, 6, 13\}$.
Hint: Put $L_1$ as a system of equations, and then put $L_2$. Solve both systems.
If I am not mistaken, $L_1=\{(x,y,z,t)\in\mathbb{R}^4 \mid z-y+x=0,\; 2t+x-5y=0\}$, $L_2=\{(x,y,z,t)\in\mathbb{R}^4 \mid 2t+3x-y-3z\}$.
Solving the system formed by the 3 equations above, we get $\langle (-1,5,6,13) \rangle$