Reference no: EM132344746

Linear Algebra Questions -

Q1. a. Find the ad-joint of the matrix.

b. Use a determinant to find an equation of the line passing through the points.

(2, 5), (6, -1)

c. Use a determinant to find an equation of the plane passing through the points.

(0, 0, 0), (2, -1, 1), (-3, 2, 5)

Q2. Determine whether S is a basis for P₃.

S = {4t - t², 5 + t³, 5 + 3t, -3t² + 2t³}

Q3. a. Find a basis for the subspace of R³ spanned by S.

S = {(1, 2, 2), (-1, 0, 0), (1, 1, 1)}

b. Find (a) a basis for the column space and (b) the rank of the matrix.

Q4. Find (a) a basis for and (b) the dimension of the solution space of the homogeneous system of linear equations.

3x₁ + 3x₂ + 15x₃ + 11x₄ = 0

x₁ - 3x₂ + x₃ + x₄  = 0

2x₁ + 3x₂ + 11x₃ + 8x₄ = 0

Q5. Find the transition matrix from B to B'.

B = {(1, 3, 2), (2, -1, 2), (5, 6, 1)},

B' = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}

Q6. a. Find the angle θ between the vetors.

u = (2, 3, 1), v = (-3, 2, 0)

b. Determine whether u and v are orthogonal, parallel, or neither.

u = (0, 3, -4), v = (1, -8, -6)

Q7. a. Show that the function does not define an inner product on R³, where u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃).

(u, v) = u₁v₁ - u₂v₂ - u₃v₃

b. Find (a) (u, v), (b) ||u||, (c) ||v||, and (d) d(u, v) for the given inner product define on Rⁿ

u = (0, -6), v = (-1, 1), (u, v) = u₁v₁ + 2u₂v₂

c. Find (a) (A, B), (b) ||A||, (c) ||b||, AND (d) d(A, B) for the matrices in M_2,2 using the inner product (A, B) = 2a₁₁b₁₁ + a₁₂b₁₂ + a₂₁b₂₁+ 2a₂₂b₂₂.

Q8. a. Find the angle θ between the vectors

u = (¼, -1), v = (2, 1),

(u, v) = 2u₁v₁ + u₂v₂

b. (a) Find proj_vu, (b) Find proj_uv, and (c) Sketch a graph of both proj_vu and proj_uv. Use the Euclidean inner product.

u = (2, -2), v = (3, 1)

Q9. Apply the Gram-Schmidt orthonormalization process to transform the given basis for Rⁿ into an orthonormal basis. Use the vectors in the order in which they are given.

B = {(0, 1, 2), (2, 0, 2), (1, 1, 1)}

Q10. Prove: (AB)^-1 = B^-1A^-1, (A^T)^-1 = (A^-1)^T.

Q11. Find uxv and show it is orthogonal to both u and v.

u =3i - j + k and v = 2i + j - k

Q12. Is T: R³ →R³, T(x, y, z) = (x+1, y+1, z+1) linear transformation.

Q13. If T: R³ → R³ and T(x) = Ax

Forfind rank and nullity of T, and determine if T is one-to-one.

Q14. Find the standard matrix for the linear transformation T T(x, y) = (4x + y, 0, 2x - 3y).

Q15. If a matrixis the matrix A' for linear transformation T: R³ → R³ relative to the standard basis. Find the matrix for T relative to the basis B' = {(1, 1, 0), (1, -1, 0), (0, 0, 1) [A' = P^-1AP].

Q16. Find the eigenvalues and a basis for each corresponding eigenspace of a matrix.

Find whether matrix A is diagonalizable. [Check P^-1AP = D]

Q17. Which of the following matrices is diagonalizable

Q18. a. Find a matrix P such that P^TAP orthogonally diagonalizes A. Verify that P^TAP gives the correct diagonal form.

b. Prove that if a symmetric matrix A has only one eigenvalue λ, then A = λI.

c. Consider the matrix below.

(a) Is A symmetric? Explain.

(b) Is A diagonalizable? Explain.

(c) Are the eigenvalues of A real? Explian.

(d) The eigenvalues of A are distinct. What are the dimensions of the corresponding eigenspaces? Explain.

(e) Is A orthogonal? Explain.

(f) For the eigenvalues of A, are the corresponding eigenvectors orthogonal? Explain.

(g) Is A orthogonally diagonalizable? Explain. 

Note - Show all paper work - neat, readable, and accurate presentation is required, each problem should be properly numerated and separated from other problems.