[Solved] Math 551 Practice Problems for Final Exam

• Attention: Textbook, notes, calculators and other electronic devices are NOT allowed during exams.
• Find a basis for the column space col(A) of the matrix

.

Based on your finding, determine the rank of A.

• Find a basis for the null space ker(A) of the matrix

.

Based on your finding, determine the rank of A.

• Find the eigenvalues of the matrix

.

4)The eigenvalues of the matrix

are ₁ = 1 and ₂ = 2. Find the corresponding eigenspaces E₁(A) and E₂(A) and their dimensions. Based on your findings, determine whether A is diagonalizable. 5) Consider the following subspace in R⁴:

1 0

W = span.

2 3

Find an orthonormal basis for W.

• Consider the following set of vectors

1 1 1

2 ^, 0 ^, 1

1 1 1

Determine whether this set of vectors forms an orthogonal basis of R³. If it does, determine whether it also forms an orthonormal basis.

• Determine a,b,c and d such that the following matrix is an orthogonal matrix

.

• Is the matrix

invertible? If yes find its inverse A¹. If no explain why. 9) In R⁴, consider the vectors

1 1 1 4

~v1 = 23 , ~v2 = 32 , ~v3 = 23 , w~ = 3028 .

4 4 4 0

Determine whether w~ belongs to the subspace V = span{~v₁, ~v₂, ~v₂}.

10) Find all solutions of the linear system

x₁ + 2x₂ + x₃ + 12x₅ = 2

x₁ + 2x₂ + 2x₃ 2x₄ + 4x₅ = 1

x₁ + 2x₂ + 5x₃ 7x₄ 18x₅ = 4