Linear Algebra Answers

Reduce the quadratic form 3x2 + 2y2 + 3z2 - 2xy - 2yz into a canonical

form using an orthogonal transformation.

Suppose T ∈ L(V ) is normal. Prove that if range T = range T*

Suppose S,TL(V) are self-adjoint. Prove that ST is self-adjoint if and only if ST=TS.

Suppose T is a normal operator on V. Suppose also that v, w "\\in" V satisfy the equations ||v||=||w||=2, Tv=3v, Tw=4w. Show that ||T(v+w)=10

Suppose n is a positive integer.Define T∈ L(Fⁿ) by T(z₁, z₂,....., z_n)=(0, z₁,..., z_n-1). Find a formula for T*(z₁, z₂,....., z_n).

Suppose (e₁,e₂,...,e_n) is an orthonormal basis of the inner product space V and v₁,v₂,...,v_n are vectors of V such that ||e_j−v_j||<1/√n. Prove that (v₁,v₂,...,v_n) is a basis of V.

Let L : R

3 −→ R3[x] be a linear transformation such that

L(1, 0, 0) = 2x + x3, L(0, 1, 0) = −2x + x2, and L(0, 0, 1) = x2 + x3

.

(a) Find a formula for L(a, b, c), where a, b, c ∈ R.

(b) Find a basis for ker L. Is L a monomorphism?

(c) Find a basis for Im L. Is L an epimorphism?

(d) Find a basis for L−1[A], where A = x^2

Find an expression of z^n which nEN