Practice for Exam 2 - Math 3130 Intro to Linear Algebra
Spring 2016 - Dr. Radu Cascaval
Exercise 1
Given the matrix

1 x x2
A =  1 2 4 ,
1 3 9
(1) Compute det(A) by the co-factor expansion method.
(2) Find all values of x such that A is not invertible.
(3) Use elementary row operations to reduce A to a row echelon form. Recompute det(A) via this
method.

Exercise 2
Given the matrix


2 0 3
A =  0 3 2 ,
−2 0 −4
decide whether A is invertible or not. If it is, compute A−1 .
Exercise 3
Solve the following linear system using Cramer’s Rule:
x+y+z = 0
x − 2y + 2z = 4
x + 2y − z = 2
Exercise 4
(a) Find the complete solution of the linear system Ax = b, where


 
1 0 1
2
2 1 1
1

 
A=
3 0 1 , and b = 0
0 −3 3
9
(b) What is the rank of the matrix A? What is the column space of A?
Exercise 5
Consider the matrix below (which happens to be the transpose of the matrix A in exercise 2(a).)


1 2 3 0
B = 0 1 0 −3
1 1 1 3
(a) Find rref(B). Is it related to rref(A) computed in exercise 2(a)?
(b) What is the rank of B? Write down a basis for the column space of B.
(c) Express the free column(s) of B as a linear combination of the columns listed in the basis found in (b).
Exercise 6
Find a basis for the subspace of R4 consisting of all vectors that are orthogonal to the following three
vectors v1 = [1, 1, 2, 0], v2 = [1, 1, 0, −1]T and v3 = [0, 0, 2, 1]T .
Exercise 7
Determine whether the following four vectors span R3 or not.
v~1 = [1, 4, 7]T ,
v~2 = [2, 5, 8]T ,
v~3 = [3, 6, 9]T ,
1
v~4 = [1, 1, 1]T
Exercise 8
Find a basis for each of the following subspaces of R4 . State the dimension of each subspace.
(1) all vectors of the form (a, a + b, 0, 0),
(2) all vectors of the form (a, b, b, b),
(3) all vectors of the form (a, a, b, c) where a + b + c = 0.
Exercise 9
Let
(1)
(2)
(3)
(4)
(5)
What
What
What
What
What


1 2 3 1 2
A =  2 4 6 2 4 .
3 6 9 1 2
is the rank of A?
is a basis for the column space of A?
are the dimensions of the row space and null space of A?
is a basis for the row space of A?
is a basis for the null space of A?
Exercise 10
Which of the following subsets W of P2 (polynomials of degree 2 or less) are subspaces of P2 ? Why or why
not?
(1) all polynomials p(x) = ax2 + bx + c where a + b = 0 and c is any real number.
(2) all polynomials p(x) = ax2 + c2 where a and c are any real numbers.
Exercise 11 Review all TRUE/FALSE questions at the end of each section 4.1-4.5, 4.7, 4.8. Below are a
few sample questions for exam 2:
Determine whether each of the following statements are TRUE or FALSE. Justify your answer for full
credit.
(1) The set of invertible 2 × 2 matrices forms a vector space with usual operations of matrix addition
and scalar multiplication.
(2) If S is any subset of a vector space V , then span(S) is a subspace of V .
(3) If S is a linearly independent set, then S does not contain the zero vector.
(4) If V = span{v1 , . . . , vn } then {v1 , . . . , vn } is a basis for V .
(5) If A and B are two n × n matrices that have the same row space, then A and B have the same
column space.
(6) The system Ax = b is consistent if b is in the column space of A.
(7) Adding one additional column to a given matrix increases its rank by one.