So I'm given a linear transformation that goes like this:
$f: \mathbb R^3 \to \mathbb R^4$ such that $f(x,y,z)=(x+z, -y+z, x+y+2z, 2y)$
and $2$ bases:
$B_1 = ((1,0,2) , (2,0,3) , (1,1,0))$
$B_2 = ((2,0,2,1) , (2,0,3,1) , (1,1,0,2) , (0,1,1,1))$
and then I'm asked to build matrix $f$ relative to the bases $B_1$ and $B_2$, and I have no idea where to go from here.
Thanks, and excuse the lack of proper formatting, I'm in a bit of a hurry here, sorry.
$B_1 = span \{u_1,u_2,u_3 \}$ and $B_2 = span \{v_1,v_2,v_3,v_4\}$
$f(u_1) = f(1,0,2) = (3,2,5,0)$
This is how $f$ maps the first vector in $B_1$ in terms of the standard basis in $\mathbb R^4$
Now we need to convert this into the basis $B_2$
$(3,2,5,0) = av_1 + bv_2 + cv_3 + dv_4$
Solve for $a,b,c,d.$
$f(u_1) = (1,-2,-1,3)$ in terms of the basis $B_2$
$f(u_2) = (5,3,8,0), f(u_3) = (1,-1,2,2)$ again in terms of the standard basis.
$f(u_2) = (3,-1,-2,3), f(u_3) = (7,13,-2,-9)$ in terms of the basis $B_2$
$f:B_1\to B_2$
$f(x_1,x_2,x_3) = (x_1 + 3x_2 + 7x_3, -2x_1 - x_2 + 13x_3, -x_1-2x_2-2x_3, 3x_1 +3x_2 - 7x_3)$