block matrix question

**transgalactic** · June 22nd 2011, 01:33 PM

there is a space V of dimention n
T:V->V
i am givem also the thing in the photo
prove that there is a basis B for such [T]_B on in the photo.
where A C D are mxm matrices.

i thought to find this basis from the eigenvectors of [T]_B
but i dont know how because each letter there is a metrices by itself

**FernandoRevilla** · June 22nd 2011, 09:47 PM

Perhaps you mean: if $T:V\to V$ is an endomorphism, $\dim V=n$ and $W$ a subspace of $V$ with $T(W)\subseteq W$ then, there exists a basis $B$ of $V$ such that $[T]_B=\begin{bmatrix}{A}&{C}\\{0}&{D}\end{bmatrix}$ . If this is the case, then choose $B=\{e_1,\ldots,e_r,e_{r+1},\ldots,e_n\}$ where $\{e_1,\ldots,e_r\}$ is a basis of $W$ . You'll easily find $[T]_B$ .

**transgalactic** · June 23rd 2011, 12:03 AM

ok so we choose some basis c1..cn .
i know that each column of $[T]_B$ is $T(c_i)_B$

what to do now, i got a very abstract thing .and i need to prove that this basis if following the conditions of the question

??

**Deveno** · June 23rd 2011, 10:33 AM

transgalactic, ask yourself this: for the basis Fernando suggested,

if we write an element of $v \in V$ as a linear combination:

$v = c_1e_1+\dots + c_re_r + c_{r+1}e_{r+1} \dots + c_ne_n$

what happens when $c_{r+1} = \dots = c_n = 0$ ?

(surely this is the case when when we have an element of W).

**transgalactic** · June 24th 2011, 01:05 PM

Originally Posted by Deveno

transgalactic, ask yourself this: for the basis Fernando suggested,

if we write an element of $v \in V$ as a linear combination:

$v = c_1e_1+\dots + c_re_r + c_{r+1}e_{r+1} \dots + c_ne_n$

what happens when $c_{r+1} = \dots = c_n = 0$ ?

(surely this is the case when when we have an element of W).

in this case v is a combination of e1 .. $e_r$
what now
stil this block matrices are hard to translate

**transgalactic** · June 28th 2011, 01:28 AM

Originally Posted by FernandoRevilla

Perhaps you mean: if $T:V\to V$ is an endomorphism, $\dim V=n$ and $W$ a subspace of $V$ with $T(W)\subseteq W$ then, there exists a basis $B$ of $V$ such that $[T]_B=\begin{bmatrix}{A}&{C}\\{0}&{D}\end{bmatrix}$ . If this is the case, then choose $B=\{e_1,\ldots,e_r,e_{r+1},\ldots,e_n\}$ where $\{e_1,\ldots,e_r\}$ is a basis of $W$ . You'll easily find $[T]_B$ .

we choose some basis of W and expand it to be the basis of V and show that the representation matrices of T on this expanding basis is the one given in thequestion.
so we are given that every T(w1)..T(wn) is a combination of W basis

for example for T(w1)=a1w1+a2w2+..+amwm+..+anwn
ALSO T(w1) is the first column
so a_m+1=a_m+2..=a_n =0 these coefficients has to be zero for the first m columns of this matrices in order for it to look we were told to.

so we just say " the coefffients a_m+1 ..a_n have to be zero in order for this basis to make such a representation matrices"
and that is the solution?

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