Description
- 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 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.
- 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 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
