MATH 2210 – Applied Linear Algebra December 9, 2015
Practice Final Exam
1. Find the standard matrix for the linear transformation T:R3→R2such that
T
1
0
0
=0
1, T
0
1
0
=1
1, T
0
0
1
=3
2.
2. True or False. If T:R2→R2rotates vectors about the origin though an angle π/10, then Tis a linear
transformation. Explain.
3. Find the eigenvalues and the eigenvectors of the matrix
5−2 3
010
6 7 −2
4. Let
A=−2 12
−1 5
Find a diagonal matrix Dand an invertible matrix Psuch that A=PDP −1. Compute A10.
5. For the following matrices Afind the basis for N ul(A), Row(A), C ol(A). What is rank(A)?
A=
12201
02110
00001
00000
.
6. If the null space of a 50 ×60 matrix Ais 40-dimensional,
(a) What is the rank of A?
(b) Null(A) is a subspace of Rn, what is n?
(c) Col(A) is a subspace of Rn, what is n?
7. Let Abe a n-by-nmatrix that satisfies A2=A. What can you say about the determinant of A?
8. Suppose a 4 ×7 matrix Ahas four pivot columns. Is Col A=R4? Is Nul A=R3? Explain.
9. Show that the set {u1,u2,u3}is an orthogonal set in R3. Then express a vector xas a linear combi-
nation of u0s, where
u1=
3
−3
0
,u2=
2
2
−1
,u3=
1
1
4
,x=
5
−3
1
.
