Linear Algebra Answers

show that the function f:R² →R² given f(x,y)=(x-y, x+y) is linear.

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.