Linear Algebra Answers

Obtain all the basic solution of the system of linear equations

Suppose b,c belongs to R. Define map T : R3 --> R2 by
T.(x,y,z)= (2x - 4y + 3z +b, 6xcxyz)
Show that T is linear if and only if b = c = 0.

Suppose T in L(V) and U is a subspace of V.
Prove that if U subset of null T, then U is invariant under T.

Define an operator T in End(F^2) by T(x,y)= (y,0) Let
U = {(x,0) | x in F}. Show that
U is invariant under T and T |U is the 0 operator on U;

Define T in L.F3/ by
T.z1; z2; z3/ D .2z2; 0; 5z3/:
Find all eigenvalues and eigenvectors of T.

Suppose T in End(F^2) is defined by
T(w,z)=(-z,w)
Find the eigenvalues and eigenvectors of T if F=R.

Suppose V is finite-dimensional, T belongs to End(V) and lambda in F . Then the
following are equivalent:
(a) lambda is an eigenvalue of T ;
(b) T -lambdaI is not injective;

Suppose, T in Hom(V,W) then prove that Nullspace of T is subspace of V

Suppose b,c belongs to R. Define map T : R3 --> R2 by
T.(x,y,z)= (2x - 4y + 3z +b, 6xcxyz)
Show that T is linear if and only if b = c = 0.

Obtain all the basic solutions of the following linear equations:
2x1 + x2 + x3 + x4 = 6
3x1 + 2x2 + x3 + 2x2 = 8