- Home
- INTERMEDIATE ALGEBRA
- Course Syllabus for Algebra I
- Mid-Plains Community College
- FRACTION OF A WHOLE NUMBER
- Systems of Linear Equations
- MATH FIELD DAY
- Course Outline for Finite Mathematics
- Calculus
- Algebra Final Examination
- Math 310 Exam #2
- Review of Trigonometric Functions
- Math 118 Practice test
- Precalculus Review
- Section 12
- Literal Equations
- Calculus Term Definitions
- Math 327A Exercise 2
- Public Key Algorithms II
- Maximizing Triangle Area
- Precalculus I Review for Midterm
- REVIEW OF A FIRST COURSE IN LINEAR ALGEBRA
- Math 6310 Homework 5
- Some Proofs of the Existence of Irrational Numbers
- ALGEBRAIC PROPERTIES OF MATRIX OPERATIONS
- Math 142 - Chapter 2 Lecture Notes
- Math 112 syllabus
- Math 371 Problem Set
- Complex Numbers,Complex Functions and Contour Integrals
- APPLICATIONS OF LINEAR EQUATIONS
- Week 4 Math
- Fractions
- Investigating Liner Equations Using Graphing Calculator
- MATH 23 FINAL EXAM REVIEW
- Algebra 1
- PYTHAGOREAN THEOREM AND DISTANCE FORMULA
- Georgia Performance Standards Framework for Mathematics - Grade 6
- Intermediate Algebra
- Introduction to Fractions
- FACTORINGS OF QUADRATIC FUNCTIONS
- Elementary Algebra Syllabus
- Description of Mathematics
- Integration Review Solutions
- College Algebra - Applications
- A Tip Sheet on GREATEST COMMON FACTOR
- Syllabus for Elementary Algebra
- College Algebra II and Analytic Geometry
- Functions
- BASIC MATHEMATICS
- Quadratic Equations
- Language Arts, Math, Science, Social Studies, Char
- Fractions and Decimals
- ON SOLUTIONS OF LINEAR EQUATIONS
- Math 35 Practice Final
- Solving Equations
- Introduction to Symbolic Computation
- Course Syllabus for Math 935
- Fractions
- Fabulous Fractions
- Archimedean Property and Distribution of Q in R
- Algebra for Calculus
- Math112 Practice Test #2
- College Algebra and Trigonometry
- ALGEBRA 1A TASKS
- Description of Mathematics
- Simplifying Expressions
- Imaginary and Complex Numbers
- Building and Teaching a Math Enhancement
- Math Problems
- Algebra of Matrices Systems of Linear Equations
- Survey of Algebra
- Approximation of irrational numbers
- More about Quadratic Functions
- Long Division
- Algebraic Properties of Matrix Operation
- MATH 101 Intermediate Algebra
- Rational Number Project
- Departmental Syllabus for Finite Mathematics
- WRITTEN HOMEWORK ASSIGNMENT
- Description of Mathematics
- Rationalize Denominators
- Math Proficiency Placement Exam
- linear Equations
- Description of Mathematics & Statistics
- Systems of Linear Equations
- Algebraic Thinking
- Study Sheets - Decimals
- An Overview of Babylonian Mathematics
- Mathematics 115 - College Algebra
- Complex Numbers,Complex Functions and Contour Integrals
- Growing Circles
- Algebra II Course Curriculum
- The Natural Logarithmic Function: Integration
- Rational Expressions
- QUANTITATIVE METHODS
- Basic Facts about Rational Funct
- Statistics
- MAT 1033 FINAL WORKSHOP REVIEW
- Measurements Significant figures
- Pre-Calculus 1
- Compositions and Inverses of Functions

# Systems of Linear Equations

1. True or False.

(a) The null space of a 4 × 6 real matrix is a subspace of R^{6}.

(b) The column space of a 4 × 6 real matrix is a subspace of R^{6}.

(c) The rank of a 6 × 4 real matrix is at most 4.

(d) The nullity of a 4 × 6 real matrix is at least 4.

(e) A vector in R^{m} is in the column space of an m × n real matrix A if the
linear

system is solvable.

(f) Any vector in the range of an m × n real matrix A can be written as a linear

combination of the column vectors of A.

(g) Suppose that an m × n real matrix A is row reduced to another matrix B by row

reductions. Then the null space of A equals the null space of B.

(h) Suppose that an m × n real matrix A is row reduced to another matrix B by row

reductions. Then the range of A equals the range of B.

(i) Suppose that an m × n real matrix A is row reduced to another matrix B by row

reductions. Then rank(A) = rank(B) and nullity(A) = nullity(B).

(j) If the column vectors of a 6 × 4 real matrix A are linearly independent, then
the null

space of A is { 0 }.

(k) If the column vectors of a 6 × 4 real matrix A are linearly independent, then
the

column space of A is R^{6}.

2. Let .

(a) Find the dimension and a basis of Span
.

(b) Find the dimension and a basis of Span . Is
in Span ? If yes,
write it as

a scalar multiple of with a specific
coefficient.

(c) Find the dimension and a basis of Span. Is
in Span ?
If yes, write

it as a linear combination of with specific
coefficients.

(d) Find the dimension and a basis of Span. Is
in Span? If

yes, write it as a linear combination of with specific
coefficients.

(e) Find the dimension and a basis of Span Is
in
Span ?

If yes, write it as a linear combination of with specific
coefficients.

Hint: The row reduction of one single matrix provides the answers to all these
questions.

3. A n × n square matrix is called a magic square if its n
row sums, n column sums, and 2

diagonal sums are all equal. For instance, is
a 3 × 3 magic square, since

8+1+6 = 3+5+7 = 4+9+2 = 8+3+4 = 1+5+9 = 6+7+2 = 8+5+2 = 6+5+4:

is another one.

Let V be the vector space of all 3 × 3 real matrices, and let M be the set of all
3 × 3

magic squares with real entries.

(a) Show that M is a subspace of V .

(b) Find the dimension and a basis of M.

**Answers:**

1. (a) Y (b) N (c) Y (d) N (e) Y (f) Y (g) Y (h) N (i) Y (j) Y (k) N

2. (a) dim = 3. Basis: .

(b) dim = 1. Basis: . Yes,
.

(c) dim = 1. Basis: . No.

(d) dim = 2. Basis: . No.

(e) dim = 3. Basis: . Yes,
.

3. (a) Verify by yourself. I'll skip here.

(b) dim(M) = 3. The following matrices form a basis of M:

Remark 1: Of course you may have different choices. That's
just fine, as long as your

basis consists of 3 different magic squares and they span M.

Remark 2: Solving this problem allows us to generate all 3 × 3 magic squares. For
the

example matrices given in the problem we have the following: