Perhaps you mean: if is an endomorphism, and a subspace of with then, there exists a basis of such that . If this is the case, then choose where is a basis of . You'll easily find .
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
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?