Linear Algebra Answers

1. Answer the following questions

                                                              →            →            →  [ 1

a) Consider the linear transformation T(x) = proj_u(x), where u =  0

                                                                                                      3 ] 

Find the matrix for T.

b) Find the matrix for the linear transformation which reflects every vector in R² across the x-axis and then rotates every vector through an angle of 𝝅/6. (Show all working)

EXERCISE 2: Find the rank and the nullity of the linear transformation S: p_1→ℝ given by 

     S(p(x)) = ∫_0^1p(x)dx.

For what values of h the vectors

⟶              ⟶                 ⟶  

u1 = [1, -3, -2]             u2 = [-1, 9, -6]              u3 = [5, -7, h]   

are linearly independent? (Show all working)

Note: The three vectors are supposed to be in a 3x1 matrix(3 rows and 1 column)

Find the value of m for which the system of equations

x - 2y + z = 0,

-2x - y + 3z = 0

y + z = m 

has only trivial solution.

Change Q= x² + 2y² + 2z² - 2xy - 2yz + zx into real canonical form and find its rank and signature.

1. For what values of h the vectors

⟶ [ 1 ⟶ [ -1 ⟶ [ 5

u₁ = -3 u₂ = 9 u₃ = -7

-2 ] -6 ] h ]

are linearly independent? (Show all working)

Find all Eigen values and corresponding Eigen vectors for the matrix A=

0 0 3

2 5 0

2 3 0

Consider the set V = R 2 . For (x1, y1),(x2, y2) ∈ R 2 and c ∈ R, define the following operations:

I. (x1, y1) + (x2, y2) = (0, y1 + y2)

II. c(x1, y1) = (0, cy1)

Is the subset a vector of R2. If not prove the axioms that makes it false.Also prove those axioms that are true.

What is grapichal method of 2x-3y=7; 3x+y=5

1. For what values of h the vectors

⟶ [1 ⟶ [-1 ⟶ [5

u₁ = -3 u₂ = 9 u₃ = -7

-2] -6] h]

are linearly independent? (Show all working)