Test 1
Linear Algebra I (201-NYC-05/01)
Winter 2011
I NSTRUCTIONS
Write your name on this question sheet and on the front of the rst page of your answer booklet.
Write complete solutions in the answer booklet, except where a question asks only for answers
Linear Algebra for Science
Wednesday, September 24th, 2014
Caroline Lefebvre
Name:_
TEST #2A
There are 38 points available to be earned on this test, but the final mark will be out of 35.
1) Consider the matrix
and its columns vectors
,
, and
. You may us
Sample Exam 2 - Linear Algebra (Instructor: J. Lucier)
Name:
Fall 2015
Score
/ 36
[2 pts] 1. Suppose u and v are nonzero parallel vectors in R3 . Select the statement that is true.
Spancfw_u, v is a plane.
cfw_u + v, u v is linearly independent.
Spancf
Sample Exam 3 - Linear Algebra
Name:
Fall 2015
Score
/0
2
1. Given A = 0
0
2. Given A =
1
2
5
1
0
1
1
4
2, find A1 .
2
write AT as a product of elementary matrices.
3. Use cofactor
expansion
(across a row or down a column of your choice) to calculate the
Linear Algebra for Science
Friday, November 28th, 2014
Caroline Lefebvre
Name:_
TEST #3A
There are 38 points available to be earned on this test, but the final mark will be out of 35.
1) Consider the matrix
a)
and its RREF
below:
[1 point] Is
a valid basi
Linear Algebra for Science
Tuesday, June 16th, 2015
Caroline Lefebvre
Name:_
TEST #1A
There are 38 points available to be earned on this test, but the final mark will be out of 35.
1) Let
and
a)
.
[1 point] Evaluate the determinant of
So
.
b) [1 point] Ev
STRATEGIES FOR PROOFS
(FOCUS ON TESTING PROPERTIES)
When showing that a property holds:
To get full marks, you must avoid examples and show
(1) What are you checking? Before you break anything down, multiply anything out, or change something for an
equiva
Linear Algebra for Science
Wednesday, September 24th, 2014
Caroline Lefebvre
Name:_
TEST #2A
There are 38 points available to be earned on this test, but the final mark will be out of 35.
1) Consider the matrix
and its columns vectors
,
, and
. You may us
Subspaces
is not a subspace:
1) If
, then show that is not a subspace by providing a
counter-example to the closed under addition property.
Closed under addition:
and
are both from .
However, their sum
is not from since it
doesnt have two identical column
Linear Algebra for Science
Tuesday, June 16th, 2015
Caroline Lefebvre
Name:_
TEST #1B
There are 38 points available to be earned on this test, but the final mark will be out of 35.
1) Consider the matrix
a)
.
[2 points] Find
.
So
b) [2 points] Use your an
Topic 1: SYSTEMS OF LINEAR EQUATIONS & MATRICES
1.1:
1.2:
1.3:
1.4:
Systems of Linear Equations & Row Reduction
Matrix Operations
The Inverse of a Matrix
Introduction to Determinants
1.1: SYSTEMS OF LINEAR EQUATIONS & ROW REDUCTION
A linear equation is an
LINEAR ALGEBRA I (NYC) TEST 3 PRACTICE
You should know.
Understand the difference between a basis and a span.
The definition of the column space as a span. The definition of the row space as a span.
The definition of the null space of as the set of all
Linear Algebra for Science
Wednesday, September 24th, 2014
Caroline Lefebvre
Name:_
TEST #1A
There are 38 points available to be earned on this test, but the final mark will be out of 35.
1) [3 points] Consider the system of linear equations
. For what va
LINEAR ALGEBRA I (NYC) FINAL REVIEW
Definitions:
Augmented matrix: A matrix made up of two matrices or more, set side-by-side. (Like
Coefficient matrix: A matrix made up of the variables coefficients in a system. (The in
Row echelon form (REF): Zeros belo
Final Examination
201-NYC-05
16 December 2014
1
1 1 c
1. (6 points) Let A = 1 c c . and b = 2 . Find the value(s) of c for which
c
c c c
(a) Ax = b has a unique solution.
(b) Ax = b has no solution.
(c) Ax = b has infinitely many solutions.
(d) Col(A) i
Final Examination
201-NYC-05
20 May 2015
1. (8 points)
Given the following
coefficient
matrix A and vector b:
6
1
1
3
3
6
1 1 3 3
A=
2 1 4 3 b = 7
5
0
1
2
3
(a)
(b)
(c)
(d)
(e)
Find the general solution to Ax = b
Find the specific solution such tha
Solutions for Test 1
Linear Algebra I (201-NYC-05/01)
2. The matrix A b is displayed and reduced below.
1
2 1 a
3
7
h
b
2
1
13 c
1 2
1
a
1
h3
3a + b
0
0
5
15
2a + c
1 2
1
a
1
h3
3a + b
0
0
0
5h + 30 17a 5b + c
1. a. The augmented matrix of the linear s
Topic 2: DETERMINANTS & VECTOR GEOMETRY
2.1:
2.2:
2.3:
2.4:
2.5:
2.6:
2.7:
The Determinant of a Matrix
Applications of the Determinant
Vectors in Space
Linear Independence
Lines, Planes, and Spaces
Projections and Distances
Spans
2.1: THE DETERMINANT OF A
Topic 3: BASES, SUBSPACES, & TRANSFORMATIONS
3.1:
3.2:
3.3:
3.4:
Bases
Subspaces
Null Space, Column Space, and Row Space
Linear Transformations
3.1: BASES AND SUBSPACES
Lets do a quick review of spans.
Span: The span of a set of vectors
in
is the set of a
Final Examination
Mathematics 201-NYC-05
Linear Algebra
21 May 2014
1. (5 points) You are given the following matrix A and vector b.
1 3
2 1
6
2 6
3 2
b = 11
A=
3 9 1 1
3
4 12
4 1
2
2
0
(a) Is
1 is a solution to the Ax = b?
2
(b) Find the ge
Linear Algebra for Science
Tuesday, February 14th, 2017
Caroline Lefebvre
Name:_
TEST #1B
There are 38 points available to be earned on this test, but the final mark will be out of 35.
1) [5 points] Recall that the graph of a quadratic function
is a parab