Linear Algebra Answers

1.Let B = (a1,a2, a3) be an ordered basis of

R3 with al= (1, 0, -1), a2= (1, 1, 1),

a3= (1, 0, 0). Write the vector v = (a, b, c) as

a linear combination of the basis vectors

from B.


2.Suppose al= (1, 0, 1), a2= (0, 1, -2) and

a3 = (-1, -1, 0) are vectors in R3and

f : R3 -> R is a linear functional such that

f(al) = 1, f(a2) = -1 and f(a3) = 3. If

a = (a, b, c) E R3, find f(a).

Given the basis {(1, - 1, 3), (0, 1, - 1),

(0, 3, - 2)} of R3, determine its dual basis.

A is a 3x4 matrix where A= (1 1 0 0 \ -1 3 0 1 \ -3 1 -2 1)

Find an orthonormal basis for the row space of the matrix using the Gram-Schmidt Process

Let λ ∈ R be an eigenvalue of an orthogonal matrix A. Show that λ = ±1.

(Hint: consider the norm of Av, where v is an eigenvector of A associated with the

eigenvalue λ.)


Also, find diagonal orthogonal matrices B, C such that 1 is an eigenvalue of B

and −1 is an eigenvalue of C.

Show that f(1, 1, 1), (1, – 1, 0), (1, 1, 0)1 is a

basis of R3 . Find a dual basis to this basis.

Let W {(x, y, z) R3: x + y + z = 0}. Check

if W is a subspace of R3 . Find a non-zero

subspace U of R3 so that W(intersection)U = (0).

Reduce the quadratic form 8x²— 4xy + 5y² to

its normal canonical form.

Consider linear operator T:C^3-->C^3 def by T(z1,z2,z3)=(z1+iz2, iz1-2z2, -iz2+z3). Check T* and check if T is self adjoined. check if T is unitary

Check whether following system of equations has a solution.

3x+2y+6z+4w=4

x+2y+2z+w=5

x+z+3w=3

Define T:R^3-->R^3 by T(x,y,z)=(-x,x-y,3x+2y+z). Check if T satisfies the polynomial (x-1)(x+1)^2.