Linear Algebra Answers

        [ 1 0 -1

3. Consider the matrix A =  0 3 0

                      -1 0 1 ]

1. Find the eigenvalues of A.
2. Find the eigenspaces corresponding to each eigenvalue from A.

2. Consider a linear transformation T: R³ → R³ defined by

  ([x         [ x + 4y +3z  

T  y     =      -5y - 4z 

    z])        5x + 10y + 7z ]

Note: T is a 3x1 matrix containing x, y, z respectively. T is equal to another matrix as shown above.

a) Find the matrix A for T

b) Find a basis for ker(T) and the dim(ker(T)). Then find dim(Im(T)), without finding a basis for Im(T). (Show all working)

c) Find a basis for Im(T)

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.