I've seen many questions about extending bases, but I still don't get the idea of extending them to a space that is NOT R4, ... etc. Here's the question:
U, W are strict-subspaces of F4.
U=span{(1,0,1,1)transpose, (2,1,-1,-3)transpose}.
W=span{(4,1,1,-1)transpose, (0,1,2,1)transpose, (1,3,-3,-8)transpose}.
Find the basis of W including (1,0,1,1)transpose and (2,1,-1,-3)transpose.
I've shown that U< W...
any help please?
Assume $U=span\{u_1, u_2\}$, $W=span\{w_1,w_2,w_3\}$. All vectors are column vectors.
Then you can write a matrix which includes the basis for $U+W$, like this: $$M = [u_1,u_2,w_1,w_2,w_3]$$
Then use Gaussian elimination on M to get an upper triangular matrix, which might be like: $$ \begin{matrix} 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0\\ \end{matrix} $$
In this case you can just add $w_2$ to $u_1,u_2$ to form your desired basis.
$$ \begin{matrix} 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 1\\ 0 & 0 & 0 & 0 & 0\\ \end{matrix} $$
In this case you can add $w_1, w_2$ or $w_3$.
Since Gaussian elimination starts from the upper left, it will always keep $u_1, u_2$ if they are linearly independent. And it shows clearly which vectors are linearly independent with respect to them. From the matrix we also see that $dim(U+W)=3$, which indicates $U\subset W$.