Linear algebra - Transformation using a Matrix

[BitterX]

Homework Statement

let $T:R^3 \rightarrow R^3$
be a Linear map,
and let $B=\left \{ (1,1,1),(1,1,0),(1,0,1) \right \}$
be a Basis

and $(1,0,0)\in kerT$$[T]_B=\begin{pmatrix}<br /> a & 0 & b\\ <br /> 3 & 2a & 1\\ <br /> 2c& b & a<br /> \end{pmatrix}$

a. find a,b,c
b. find a Basis for ImT

Homework Equations

The Attempt at a Solution

as for a.

I think that what we need to do is find a general vector (x,y,z)
and express (1,0,0) thorugh it
if we multiply the matrix by what we got we will have to get (0,0,0) (because (1,0,0) is in the kernel)
but I'm not sure how to express (1,0,0).
I think the vector (1,0,0) in the basis B is -1(1,1,1)+1(1,1,0)+1(1,0,1) and so is
(-x+y+z,0,0)

and for the matrix we have
-a+0+b=0
-3+2a+1=0
-2c+b+a=0

from the second equation:
a= 1
from the first
b=1
from the third
c=1

Am I right?

 

[lanedance]

yeah look good calling the natural basis E, (1,0,0) in the B basis is as follows
$$(1,0,0)^T_E = (-1,1,1)^T_B$$

And as a check in E
$$-1\textbf{b}_1+1\textbf{b}_2+1\textbf{b}_3 = -1(1,1,1)^T+1(1,1,0)^T+1(1,0,1)^T=(-1+1+1,-1+1+0,-1+0+1)^T =(1,0,0)^T$$