MTH 341 Starters
Class info:
Class: MTH 341-020
CRN: 52729
Time: MWF 1:00-1:50
Room: LINC 302
Textbook: A First Course in Linear Algebra, Version 2017 Revision A, by K. Kuttler.
https://tinyurl.com/yc8lxz2u
Instructor: Dr. Roger Roybal (he/him) (preferred name: Roger or Dr. Roybal)
email: roybalr@oregonstate.edu
Office: KIDD 290
Hours: MW 4:30-6:00pm
Graduate Teaching Assistant: Madison Phelps
email: phelpmad@oregonstate.edu
Office and Hours:
More Info:
Class Topics
ˆ Systems of equations
ˆ Matrices
ˆ Determinants
ˆ Vector spaces (Rn )
ˆ Linear independence, span, basis, and dimension
ˆ Linear transformations, connection with matrices, Rank-Nullity theorem
ˆ Eigenvectors/eigenvalues - spectral theory
Section 1.1: Systems of Equations: Geometry
A linear equation from intro to Algebra:
ax = b,
where a, b ∈ R.
Q: How many solutions does this equation have?
a ̸= 0
a = 0 and b = 0
Now consider the equation in variables x and y:
ax + by = c,
with a, b, c ∈ R and at least one of a or b is nonzero. We can graph this:
a = 0 and b ̸= 0
This equation is sometimes called a 1×2 system of equations: 1 equation with 2 variables.
Note: This equation ax + by = c (with at least one of a or b being nonzero) has one
degree of freedom. We can freely choose one variable, but then the second is completely
determined by the first chosen.
Given a system of equations
(
ax + by
dx + ey
=c
= f,
with a, b, c, d, e, f ∈ R, a solution to this system is an ordered pair (x, y) which satisfies both
equations simultaneously.
Graphing:
One solution
No solutions
This system is often called a 2 × 2 system.
Infinite solutions
A 3 × 3 system – 3 equations with 3 variables:
ax + by + cz = d
ex + f y + gz = h
ix + jy + kz = l
Each equation describes a plane individually. How do they intersect?
Section 1.2: Systems of Equations
A system of linear equations is a list of equations
a11 x1 + a12 x2 + · · · + a1n xn = b1
a21 x1 + a22 x2 + · · · + a2n xn = b2
..
.
am1 x1 + am2 xn + · · · + amn xn = bm ,
where the aij are scalars, xj are variables. This is a m × n system: m equations with n
variables.
Equivalently we can write this as
n
X
j=1
aij xj = bi
f or
i = 1, 2, . . . , m.
Elementary Operations
Let’s solve the system
x + 2y = 3
5x − 3y = 4.
Eliminating variables?
Three elementary operations for solving a linear system:
Convenient notation (due to Cayley): instead of writing all the variables, we write the
coefficients in an array called an augmented matrix. We use a vertical line to separate the
coefficients on the left from the constants on the right.
A m × n matrix has m rows and n columns. Note that the augmented matrix for an
m × n system of linear equations will have n + 1 columns, with the last column consisting
of the constants to the right of the equals sign.
Example 1. Write the augmented matrix corresponding to the system
(
x + 2y
=3
5x − 3y = 4.
Gaussian Elimination
Since a matrix keeps track of variables by positions of entries, we can do row operations on
a matrix, and we’ve eliminated a variable in an equation when the corresponding entry in
the matrix is zero.
Note: Elementary operations on a system of equations ←→ matrix row operations
Row Echelon Form (ref ) and Reduced Row Echelon Form (rref )
We want to row reduce a matrix to a form where we can read off a solution to a system of
linear equations. A matrix is in row echelon form if
1. All zero rows are below any nonzero rows.
2. Each leading (first nonzero) entry of a row is to the right of leading entries of rows
above it.
3. Each leading entry of a row is 1.
Additionally, a matrix is in reduced row echelon form if it is in row echelon form, plus
4. All entries above and below a leading entry are 0.
Example 2 (Nonexamples).
2 1 9
0 1 −3


1 0 6
0 0 0
0 1 0
0 1 | −1
1 0 | 2
1 0 | 3
0 −1 | −2
Example 3. Which of these matrices are in row echelon form and reduced row echelon form
(Note that a matrix in rref is also automatically ref)?
1 1 9
1 0 | −1
0 1 −3
0 1 | 2


1 0 6
0 1 0
0 0 0
1 2 | 3
0 1 | −2