Linear Algebra Answers

2x + y = 7

x - 2y = 1

A. Write the equation in matrix form.

B. Determine the inverse of the matrix

C. Hence solve the equations.

D. x and y are matrices

"X=\\begin{bmatrix}\n 1 & 5 \\\\\n 3 & 7\n\\end{bmatrix} \n \n\n Y=\\begin{bmatrix}\n 3 & 4 \\\\\n 2 & 1\n\\end{bmatrix}"

Evaluate X² + Y

        [ 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.