Trouble understanding linear transformations in this context

[Eclair_de_XII]

Homework Statement

"Show that every subspace of $ℝ^n$ is the set of solutions to a homogeneous system of linear equations. (Hint: If a subspace $W$ consists of only the zero vector or is all of $ℝ^n$, $W$ is the set of solutions to $IX=0$ or $0_vX=0$, respectively.

Assume $W$ is not one of these two subspaces. Let $β=\{v_1,...,v_k\}$ for $W$. Extend this basis to $β`=β∪\{v_{k+1},...,v_n\}$ to span all of $ℝ^n$. Let $T: ℝ^n → ℝ^n$ be the linear transformation so that $T(v_1)=T(v_2)=...=T(v_k)=0$ and $T(v_j)=I(v_j)$ for $k+1≤j≤n$, where $I(v)$ is the identity function. Now use $T$ to obtain a matrix A so that $W$ is the set of solutions to the homogeneous system $AX=0_v$.)"

Homework Equations

$T_\alpha=PT_βP^{-1}$

where $P$ is the transition matrix from basis $β$ to basis $α$.

The Attempt at a Solution

$T(c_1v_1+...+c_kv_k)=T(c_1v_1)+...+T(c_kv_k)=c_1T(v_1)+...+c_kT(v_k)=0_v$
$T(c_{k+1}v_{k+1}+...+c_nv_n)=c_{k+1}T(v_{k+1})+...+c_nT(v_n)=c_{k+1}v_{k+1}+...+c_nv_n=0_v$

I honestly don't know what I'm doing here, and if anyone would like to provide feedback on what I'm doing wrong, or what I should actually be doing instead of this, that would be much appreciated. Do I have to left-multiply the $n×n$ transformation matrix by a column vector whose entries consist of the constants $c_i$, proving that the product is the zero vector, and that it solves the homogeneous system? $\left[T(v)\right]\left[c_i\right]=\left[0_v\right]$. Something like that, maybe? Sorry, and thanks.

 

[pasmith]

Consider the diagonal matrix \Lambda = \mathrm{diag}(\lambda_1, \dots, \lambda_n) and the matrix P = \begin{pmatrix} v_1 & v_2 & \dots & v_n\end{pmatrix} where \{v_i : i = 1, 2, \dots, n\} is a basis.

What is P\Lambda P^{-1} v_i?

 

[jambaugh]

I would suggest firstly that you write the matrix form of the equation in the first constructed basis where you have a basis of the subspace W extended to the rest of the space. Try this with say 3 dimensions and W the x-y plane. Work out the details of the concrete example and then see if you can understand the generalization.

Once you have the system of equations in the original basis you transform to the arbitrary basis with the T matrix.

 

[Eclair_de_XII]

What is $P\Lambda P^{-1} v_i$?

It looks a solution to $\Lambda v_i$ using a different basis, I think

Try this with say 3 dimensions and W the x-y plane.

Okay, so...

Let $W⊆ℝ^2$ be spanned by $\beta=\{b_1,b_2\}$. Then let $\beta'=\beta∪\{y\}$.

Let $P=\left[b_1|b_2|y\right] = \begin{pmatrix} a & d & x \\ b & e & y \\ c & f & z \end{pmatrix}$.

Then I apply the transformation matrix,

$T=\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$

So... $PT=\begin{pmatrix} 0 & 0 & x \\ 0 & 0 & y \\ 0 & 0 & z \\ \end{pmatrix}$