# The Basis of $U+W$.and $U \cap W$.

so i tryed to get the Basis for $U + W$ and $U \cap W$.

$U=\operatorname{span}\{(1,0,2,-1),(0,1,3,1)\}$

$W=\operatorname{span}\{(1,1,-1,2),(0,1,9,-1)\}$

thats all in $\mathbb{R}^4$

so first i get for $U + W$

$$\begin{cases} 1\alpha_1 + 0\alpha_2 + 1\alpha_3 + 0\alpha_4 = 0 \\0\alpha_1 + 1\alpha_2 + 1\alpha_3 + 1\alpha_4 = 0 \\2\alpha_1 + 3\alpha_2 + -1\alpha_3 +9\alpha_4 = 0 \\-1\alpha_1 + 1\alpha_2 + 2\alpha_3 + -1\alpha_4 = 0 \end{cases}$$

and if i get solve this system what ive to do?

so now i've to get the $U \cap W$.

for that i was confused what i have to do. The other one was only to calculate with $+$ if im not wrong.

• Please, if you are ok, you can accept the answer and set it as solved. Thanks!
– user
Commented Jan 17, 2018 at 22:58

If they are independent a basis for $U+W$ is any basis of $\mathbb{R^4}$ and the dimension of $U \cap W$ wold be $0$.
• Thank you. so i got:$$\begin{cases} 1\alpha_1 + 0\alpha_2 + 1\alpha_3 + 0\alpha_4 = 0 \\0\alpha_1 + 1\alpha_2 + 1\alpha_3 + 1\alpha_4 = 0 \\0\alpha_1 + 0\alpha_2 + -6\alpha_3 +6\alpha_4 = 0 \\0\alpha_1 + 0\alpha_2 + 0\alpha_3 + 0\alpha_4 = 0 \end{cases}$$ so $\alpha_1 = -t$$__$$\alpha_2 = -2t$$__$$\alpha_3 = t$$__$$\alpha_4 = t$ Commented Dec 5, 2017 at 11:30