- #1
Terrell
- 317
- 26
Homework Statement
Let ##W_1=\langle (1,2,3,6),(4,-1,3,6)(5,1,6,12))\rangle## and ##W_2=\langle (1,-1,1,1),(2,-1,4,5)\rangle## be subspaces of ##\Bbb{R}^4##. Find the bases for ##W_1\cap W_2## and ##W_1+W_2##.
Homework Equations
##\alpha(1,2,3,6)+\beta(4,-1,3,6)=\gamma(1,-1,1,1)+\delta(2,-1,4,5)##
The Attempt at a Solution
I began by determining if the vectors in ##W_1## are linearly independent, then found out that ##dim(W_1)=2##, then likewise ##dim(W_2)=2## and ##W_2## is linearly independent. Now would it be correct to put the basis vectors of ##W_1## & ##W_2## in a matrix to find ##W_1\cap W_2##? Since if I let ##\{(1,2,3,6),(4,-1,3,6)\}## and ##\{(1,-1,1,1),(2,-1,4,5)\}## be bases for ##W_1## & ##W_2##, respectively, then ##W_1\cap W_2## can be expressed in the following equation ##\alpha(1,2,3,6)+\beta(4,-1,3,6)=\gamma(1,-1,1,1)+\delta(2,-1,4,5)## which can be solved by performing r.r.e.f. on the following matrix
\begin{bmatrix}
1 & 4 & 1 & 2 \\
2 & -1 & -11 & -1\\
3 & 3 & 1 & 4\\
6 & 6 & 1 & 5\\
\end{bmatrix}
I then obtained
\begin{bmatrix}
1 & 0 & 0 & 0.\overline{7} \\
0 & 1 & 0 & -0.\overline{4}\\
0 & 0 & 1 & 3\\
0 & 0 & 0 & 0\\
\end{bmatrix}
Does this mean that ##W_1\cap W_2## is a point/vector in 4-D space?
Furthermore, a basis for ##W_1+W_2## is the set ##\{(1,2,3,6),(4,-1,3,6),(1,-1,1,1)\}##. I need help since my book didn't provide a solution.