Linear Algebra Answers

T :ℝ2 → ℝ2 as, 𝑇 𝑥, 𝑦 = (1, 𝑦) ;is it a linear transformation?

26. Reduce the Quadratic Form (Q.F.) 2x

2 + 5y

2 + 3z

2 + 4xy to canonical form by an

orthogonal transformation. Also find its nature, rank, index and signature of the Q.F.

Diagonalize the matrix





10 −2 −5

−2 2 3

−5 3 5





If 2 is one of the eigenvalue of





−2 2 −3

2 1 −6

−1 −2 0



 then find the other two eigenvalues.

2. Use Cayley-Hamilton theorem to find A6 − 5A5 + 8A4 − 2A3 − 9A2 + 31A − 36I,

when A=





1 0 3

2 1 −1

1 −1 1





10.) Consider the linear equation 2a + 3b = 4

Is (a; b) = ( 12 ; 1) a solution to the equation? Motivate your answer.

11.) Look up what is meant by a system of linear equations.

A known fact of solutions of systems of linear equations is that only one the following options can hold :

(a) No solution possible

(b) A unique solution can be found

(c) The system has infinite solutions.

Consider that two straight lines form a linear system.

Interpret what happens geometrically to the straight lines to get each case of the solution types given above.

12.) Look up the concept of a homogeneous linear system.

Only two solution types of the three mentioned solution types above are possible. Which one can never happen and why.

4.) True or False : 3Z = Z + Z + Z when Z is a matrix.

􏰀1 2􏰁 􏰀a􏰁

5.) Let X = 3 4 ; E = b

Find each of the following. If the operation cannot be done : state undefined operation.

a) XE

b) EX

c) XT X where XT stands for the transpose of X

10.) Consider the linear equation 2a + 3b = 4

Is (a; b) = ( 1/2 ; 1) a solution to the equation? Motivate your answer.

11.) Look up what is meant by a system of linear equations.

A known fact of solutions of systems of linear equations is that only one the following options can hold :

(a) No solution possible

(b) A unique solution can be found

(c) The system has infinite solutions.

Consider that two straight lines form a linear system.

Interpret what happens geometrically to the straight lines to get each case of the solution types given above.

12.) Look up the concept of a homogeneous linear system.

Only two solution types of the three mentioned solution types above are possible. Which one can never happen and why.

Show that if be the Eigenvalues of the matrix, then has the Eigenvalues

.

λ1, λ2, λ3, . . . λn A An

λn

1 , λ n

2 , λn

3 . . . λn

n