Recent content by bernoli123

1. Matrix and basis

Consider a) f1=1, f2=sinx , f3=cosx b) f1=1, f2=ex , f3=e2x c)f1=e2x , f2=xe2x f3=x2e2x in each part B={f1,f2,f3} is a basis for a subspace V of the vector space. Find the matrix with respect to B of the differentiation operator D:V→V
2. Inner product of polynomials

[-1]int[1]P(x)Q(x)dx P,Q\inS verify that this is an inner product.
3. Skew symmetric matrix

what about thinking of rank-nullity theory since the dimension of this skew-symmetric matrix=n(n-1)/2 but how to calculate the dim of the AX=0
4. Skew symmetric matrix

how can we prove that the rank of skew symmetric matrix is even i could prove it by induction is there another way
5. Isomorphic linear space

two linear spaces S and S1 over F are isomorphic if and only if there is a one-to-one correspondence x↔ x1 between the elements x \in S and x1 \in S1 such that if x ↔ x1 and y ↔ y1 then x+y ↔ x1+y1 and ax ↔ ax1 (y \in S , y1 \in S1, a \in F). prove that two finite -dimensional spaces are...
6. Denote standard inner product

if A \in C nxn,show that (x,Ay)=0 for all x,y \in C[n], then A=0 (x,Ay) denote standard inner product on C[n]
7. Rank of matrix

check that, for any nxn matrices A,B then rank(AB) (> or =) rank A +rank(B)-n
8. Show that if all the row sums of a matrix A belong to C (nxm) are

thank you very much
9. Show that if all the row sums of a matrix A belong to C (nxm) are

show that if all the row sums of a matrix A belong to C (nxm) are zeroes, then A is singular. Hint. Observe that Ax=0 for x=[1 1 ....1]T