Linear Algebra Answers

a) Let scalar be real numbers. Show that the set of real-valued, continuous functions on

the closed interval [0 ,1] forms a vector space.

b) Also show that those functions in part (a) for which all nth derivatives exist for

Prove that a set of n linearly independent vectors in a n–dimensional

vector space V span V

Qs: 04 Let T(x, y, z, w) = ( 3y -4z, x +2y+4z-w, 7z, -y-w)


Write the standard matrix for T. Check if T is one-to-one and onto.

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;