Practice Problems for Final Exam
• Attention: Textbook, notes, calculators and other electronic devices are NOT allowed during exams.
1) Find a basis for the column space col(A) of the matrix
.
Based on your finding, determine the rank of A.
2) Find a basis for the null space ker(A) of the matrix
.
Based on your finding, determine the rank of A.
3) Find the eigenvalues of the matrix
.
4)The eigenvalues of the matrix
are λ1 = −1 and λ2 = 2. Find the corresponding eigenspaces Eλ1(A) and Eλ2(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.
6) 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.
7) Determine a,b,c and d such that the following matrix is an orthogonal matrix
.
8) Is the matrix
invertible? If yes find its inverse A−1. 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
