1. ## Basis

Let $U_1$ and $U_2$ be the subspaces of $R^4$ defined by

$U_1 = (x = (x_1,x_2,x_3,x_4)|x_1 + 2x_2 - x_3 - x_4 = 0)$

and

$U_2 = (x = (x_1,x_2,x_3,x_4)|x_1 - x_2 + x_3 + x_4 = 0)$

Find basis for $U_1$ and $U_2$ and $U_1 \cap U_2$

Now i can find basis for $U_1$ and $U_2$ fairly easily tho they were different from the ones given in the solutions. Problem is i have no idea how to calculate $U_1 \cap U_2$. This is the solution given.

(-2, 1, 0, 0), (1, 0, 1, 0) and (1, 0, 0, 1) form a basis for $U_1$

(1, 1, 0, 0), (-1, 0, 1, 0) and (-1, 0, 0, 1) form a basis for $U_2$

(-1, 2, 3, 0), (-1, 2, 0, 3) form a basis for $U_1 \cap U_2$
can someone explain this? It just says clearly this is the basis in the solution... Maybe clear for someone whos been studying it for 30+ years... I can sorta see where the -1 and 2 come from but not the 3's. Any help please?

Thanks

Let $U_1$ and $U_2$ be the subspaces of $R^4$ defined by

$U_1 = (x = (x_1,x_2,x_3,x_4)|x_1 + 2x_2 - x_3 - x_4 = 0)$

and

$U_2 = (x = (x_1,x_2,x_3,x_4)|x_1 - x_2 + x_3 + x_4 = 0)$

Find basis for $U_1$ and $U_2$ and $U_1 \cap U_2$

Now i can find basis for $U_1$ and $U_2$ fairly easily tho they were different from the ones given in the solutions. Problem is i have no idea how to calculate $U_1 \cap U_2$. This is the solution given.

(-2, 1, 0, 0), (1, 0, 1, 0) and (1, 0, 0, 1) form a basis for $U_1$

(1, 1, 0, 0), (-1, 0, 1, 0) and (-1, 0, 0, 1) form a basis for $U_2$

(-1, 2, 3, 0), (-1, 2, 0, 3) form a basis for $U_1 \cap U_2$
can someone explain this?
If $(x_1,x_2,x_3,x_4)\in U_1\cap U_2$ then $x_1 + 2x_2 - x_3 - x_4 = 0$ and $x_1 - x_2 + x_3 + x_4 = 0$. The way to find solutions to a set of equations like that is to form the matrix of coefficients and row-reduce it.

In this case, the matrix is $\begin{bmatrix}1&2&-1&-1\\1&-1&1&1\end{bmatrix}$. To row-reduce it, subtract the top row from the bottom row, getting $\begin{bmatrix}1&2&-1&-1\\0&3&-2&-2\end{bmatrix}$. That should help you to see where the 2s and 3s come from.

3. sorry but i still have no idea. Can someone please just explain this fully. I have an exam on linear algebra in 2 days and i really dont wanna waste hours trying to figure out something thats only gonna get me a mark or two in the exam if its even in it. And i cant just ignore it cos it'll bug me to the point ill get nothing else done!