Linear Algebra Answers

Let a quadratic form have the expression x2+y2+2z2+2xy+3xz with respect to the
standard basis B1 = f(1;0;0); (0;1;0); (0;0;1)g. Find its expression with respect to the
basis B2 = f(1;1;1); (0;1;0); (0;1;1)g

If A and B are two matrices of same order and rank (A)=rank (B)=n ,
then rank (A+B)=n , for n>=1 .

Let f:C3 to C be defined as f(z)=(z1-z2)-i(2z1+z2+z3), where z=(z1,z2,z3) belongs to C3.
Find aw belongs to C3 such that f(z)=<z,w>, where <,> is the standard inner product on C3.

If V is an eigenvector of an n*n invertible matrix A, then V is also an eigenvector of the matrix A2.

If { V1,V2,V3 } is a linearly independent set in R3 , then so is { V1+V2-2V3 , V1-2V2+V3 , -2V1+V2+V3 }.

Find a basis of the subspace of R3 of the solution of the equation X+Y+Z=0 .

Determine whether the following subsets are subspaces of the given vector spaces:
(1) S={A belongs to Mn(C) | A is skew-Hermition} , vector space Mn(C) over C.
(2) W={ f(x) belongs F[x] f(0)=0 }, vector space F[x] over F.

Let (V,<,>) be an inner product space over C and T belongs to A(V) . Prove that if <Tx , Ty>=<x,y> for all x,y belongs to V , then T is unitary.

Find the rank of the quadratic form 2x2+2y2+3z2-xy+4yz +5xz .

Find the dual basis of the basis e1=(1,1,2) , e2=(1,0,1) , e3=(2,1,0) of the vector space R3 over R.