Linear Algebra Answers

Let f be the bilinear form on R² defined by

f[(x₁, x₂), (y₁, y₂)]= 3x₁y₁-2x₁y₂+4x₂y₁-x₂y₂

then find the matrix A of f in the basis {u₁(1, 1)=u₂(1, 2)}

Express A^5-A^4+A^2-4I as linear polynomial in A where A=metrix 3 1

-2 2

x+y+z=9

2x-3y+4z=13

3x+4y+5z=40

Check whether the set 

S = {(al, a2, ..., an) E Rn I a1= 1 + a2} 

is a subspace of Rn or not. 

Let T be the linear operator on R² defined by

T(x, y) = (−y, x)

i. What is the matrix of T in the standard ordered basis for R² ?

ii. What is the matrix of T in the ordered basis B = {α₁, α₂}, where α₁ = (1, 2) and α₂ = (1, −1)?

iii. Prove that for every real number c the operator (T − cI) is invertible.

Determine the range, kernel, rank and nullity of the given matrix,

A = [2 3 1 2 0

0 3 −1 2 1

1 -3 2 4 3

2 3 0 3 0]

Let F be the field of complex numbers and let T be the function from F

3

into F

3 defined by

T(x1, x2, x3) = (x1 − x2 + 2x3 ,2x1 + x2 , − x1 − 2 x2 + 2x3 )

i. Verify that T is a linear transformation.

ii. If (a, b, c) is a vector in F

3, what are the conditions on a, b and c that the vector bein the range of T? What is the rank of T?

iii. What are the conditions on a, b and c that the vector (a, b, c) be in the null space of T? What is the nullity of T?

Let T be the linear operator on R

3 defined by

T(x1, x2, x3) = (3x1, x1 − x2 , 2x1 + x2 + x3 )

Is T invertible? If so, find a rule for T −1 like the one which defines T.

 Let (V, <, >) be an inner product space and

let T belongs to A(V). Prove that the following 

conditions are equivalent. 

(i) T*T =I

(ii) <Tx, Ty> = <x,y> for all x, yEV 

(iii) ||Tx||=||x||for all xEV. 

using the following table, find f(x) as a polynomial in x:

x: – 1 0 3 6 7

f(x): 3 – 6 39 822 1611.