$V $ is Vector space , $W$ and $T$ Subspace.
$W=\{(a,b,c)| 2a-b+3c=0\}$
$T=\{(x,y,z)| x+ y-z=0\}$
$W+T$ is total subspace.
How do we find base and size of $ W+T $subspace ?
How to find the Linear algebra total subspace base and size
2
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calculus
linear-algebra
vector-spaces
2 Answers
0
Hint. Verify that:
(a)$\quad B_W=\{u_1=(1,2,0),u_2=(0,3,1)\}$ is a basis of $W$.
(b)$\quad B_T=\{v_1=(-1,1,0),v_2=(1,0,1)\}$ a basis of $T$.
(c)$\quad\text{rank }[u_1,u_2,v_1,v_2]=3$.
So, $W+T=V.$
0
$W=\operatorname{span}\{(2,-1,3)^T\}^\perp$ and $T=\operatorname{span}\{(1,1,-1)^T\}^\perp$, so each is two-dimensional. Hence, their sum will span $V$ unless $W=T$. It’s pretty clear from inspection that they’re not the same subspace, but you can verify this by computing the dimension of their intersection, which is the dimension of the solution space of the system $$\begin{align}2x-y+3z&=0\\x+y-z&=0.\end{align}$$ or equivalently, row-reduce the coefficient matrix of this system to find its nullity.
If $W+T=V$, you can, of course, simply take the standard basis as your basis. If not, a basis for either of $W$ or $T$ will do, which I expect you already know how to find.