# Thread: Linear Transformation and Matrix Representations

1. ## Linear Transformation and Matrix Representations

Let V be a n-dimensional vector space, and let $T$ be a linear operator on $V$
Suppose that $W$ is a $T$-invariant subspace of $V$, $dim(W)=k$
Show that there exists a basis $\alpha$ for $V$ such that $[T]_{\alpha}$ has the form

$$\left( {\begin{array}{cc} A & B \\ O & C \\ \end{array} } \right)$$

where $A$ is a $k*k$ matrix and $O$ is the $(n-k)*k$ zero matrix

Request
The question itself confuses me. I am not sure what the question wants

If $\alpha$ is a basis for $V$, then $\alpha$ has n vectors and
$[T]_{\alpha}$ is a $n*n$ matrix

So what is this question asking us to reduce the matrix to a $2*2$ matrix with the entries themselves being matrices?

2. You can choose a base for $W$ and complete it to get a base of $V$.
The main interest is to write $T$ in a simpler way.

3. Originally Posted by I-Think
Let V be a n-dimensional vector space, and let $T$ be a linear operator on $V$
Suppose that $W$ is a $T$-invariant subspace of $V$, $dim(W)=k$
Show that there exists a basis $\alpha$ for $V$ such that $[T]_{\alpha}$ has the form

$$\left( {\begin{array}{cc} A & B \\ O & C \\ \end{array} } \right)$$

where $A$ is a $k*k$ matrix and $O$ is the $(n-k)*k$ zero matrix

Request
The question itself confuses me. I am not sure what the question wants

If $\alpha$ is a basis for $V$, then $\alpha$ has n vectors and
$[T]_{\alpha}$ is a $n*n$ matrix

So what is this question asking us to reduce the matrix to a $2*2$ matrix with the entries themselves being matrices?
That's one way of looking at it but it is really just saying that the matrix has the left k columns of the bottom n-k rows all 0s. Use girdav's suggestion- select an ordered basis for V such that the first k vectors form a basis for W. Remember that the matrix representation of a linear transformation, in a given basis, has columns that are the coefficients of the vector you get by applying the linear transformation to the basis vectors. Since the first k basis vectors are in W and W is T-invariant, what will those first k columns look like?