# Basis

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• May 12th 2008, 12:29 PM
Deadstar
Basis
Can someone explain how to get this answer please.

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

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

and

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

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

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

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

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

(-1, 2, 3, 0), (-1, 2, 0, 3) form a basis for $\displaystyle U_1 \cap U_2$
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
• May 13th 2008, 12:04 AM
Opalg
Quote:

Originally Posted by Deadstar
Can someone explain how to get this answer please.

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

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

and

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

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

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

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

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

(-1, 2, 3, 0), (-1, 2, 0, 3) form a basis for $\displaystyle U_1 \cap U_2$
can someone explain this?

If $\displaystyle (x_1,x_2,x_3,x_4)\in U_1\cap U_2$ then $\displaystyle x_1 + 2x_2 - x_3 - x_4 = 0$ and $\displaystyle x_1 - x_2 + x_3 + x_4 = 0$. 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 $\displaystyle \begin{bmatrix}1&2&-1&-1\\1&-1&1&1\end{bmatrix}$. To row-reduce it, subtract the top row from the bottom row, getting $\displaystyle \begin{bmatrix}1&2&-1&-1\\0&3&-2&-2\end{bmatrix}$. That should help you to see where the 2s and 3s come from.
• May 13th 2008, 06:23 AM
Deadstar
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!