Linear Algebra - Lecture 1 - Fall 2026

Overview

Nathanson math lectures introduces linear algebra through linear equations and their solution sets, defining vectors in R^n, homogeneous versus inhomogeneous equations, and the result that every nonzero linear equation over the reals has solutions. The lecture then establishes coordinate-wise vector addition and subtraction, scalar multiplication, and representations of solution sets as scalar multiples or as a particular solution plus variable multiples of direction vectors.

Key takeaways

A nonzero linear equation in n real variables always has a solution: it has exactly one solution when n = 1 and infinitely many when n ≥ 2.
For a₁x₁ + ⋯ + aₙxₙ = b, choosing any nonzero coefficient aⱼ lets you assign the other n − 1 variables freely and solve for xⱼ.
The solution set of 3x − 2y − 5z = 17 can be expressed as one particular solution, (0, 0, −7/5), plus arbitrary multiples of two direction vectors.
A solution set can have different-looking parameterizations—for example, solving 3x − 2y − 5z = 17 for z or for y—while still representing exactly the same vectors.
In this course, vector addition and subtraction are coordinate-wise, scalar multiplication scales every coordinate, and vectors must be written as columns for later matrix calculations.

Chapters

0:00 Math 313 Format, Exams, and Course Grading

Math 313 lectures meet on Zoom and are recorded for posting to the Nathanson math lectures YouTube channel.
The midterm and final are in person at 11:00, with no phones, calculators, computers, or other electronic devices permitted.
Grades consist of homework at 30%, the midterm at 30%, and the final at 40%.

4:00 Brightpace Notes, Homework, and Handwritten Exam Skills

The first chapter’s 47 pages of lecture notes are on Brightpace and are expected to support the first two or three weeks of class.
Homework problems come from the notes; the first assignment was to be posted after class, with problems including exercises on page 20.
Home assignments may use tools such as calculators, computers, Maple, or AI, but students must solve problems by hand during exams.

5:03 The Equation 4x + 7y = 0 Has Many Solutions

For 4x + 7y = 0, Nathanson math lectures verifies that (7, −4) and (0, 0) are solutions.
Choosing x = 3 forces y = −12/7, illustrating how one chosen coordinate determines another.
Checking a proposed pair by substitution—here, 4(3) + 7(−12/7) = 0—is emphasized as a basic mathematical habit.

8:30 All Solutions of 4x + 7y = 0 as Vectors

Instead of finding one answer, the goal is to describe every solution as a two-coordinate vector (x, y).
Solving for y gives y = −(4/7)x, so the complete solution set is {(x, −(4/7)x) : x ∈ R}.
The notation introduces R as the real numbers and uses curly braces to denote a set.

13:02 Coordinates, Vectors, and the Spaces R², R³, and Rⁿ

A vector in R² has two real coordinates, while a vector in R³ has three.
For any positive integer n, Rⁿ denotes the set of column vectors with n real coordinates.
Linear algebra treats these collections of vectors as central objects, even when starting from a simple equation.

16:13 Homogeneous and Inhomogeneous Linear Equations

The equation 4x + 7y = 0 is homogeneous because its right-hand-side scalar is zero.
The equation 4x + 7y = 5 is inhomogeneous because its right-hand side is nonzero.
For 4x + 7y = 5, both (0, 5/7) and (3, −1) are example solutions.

20:07 The Solution Space of 4x + 7y = 5

A solution space is the set of all vectors that solve a linear equation or system.
Choosing any real x and solving for y gives y = −(4/7)x + 5/7.
The full solution space is {(x, −(4/7)x + 5/7) : x ∈ R}, a subset of R².

23:51 General Form of a Linear Equation in n Variables

A linear equation in n variables has the form a₁x₁ + a₂x₂ + ⋯ + aₙxₙ = b.
It is called nonzero when at least one coefficient aᵢ is nonzero; otherwise its variable terms all vanish.
A solution is a vector in Rⁿ whose coordinates satisfy the equation, and the solution space collects all such vectors.

27:45 Solving 3x − 2y − 5z = 0 in R³

The vector (1, −1, 1) solves 3x − 2y − 5z = 0 because 3 + 2 − 5 = 0.
Choosing any x and y determines z = (3/5)x − (2/5)y.
For example, x = −8 and y = 5 give z = −34/5; substitution confirms the equation equals zero.

33:03 A Particular Solution to 3x − 2y − 5z = 17

The right-hand side 17 makes 3x − 2y − 5z = 17 inhomogeneous.
Setting x = −1 and y = 5/2 gives z = −5, and substitution verifies 17.
For arbitrary x and y, solving for z yields z = −17/5 + (3/5)x − (2/5)y.

38:36 Free Choices and Equivalent Descriptions of One Solution Set

Any two coordinates can be chosen freely in 3x − 2y − 5z = 17, with the remaining coordinate determined by the equation.
Solving for y instead gives y = −17/2 + (3/2)x − (5/2)z for arbitrary x and z.
These parameterizations look different but describe exactly the same set of vectors in R³.

43:36 Why Mathematical Claims Require Proof

Nathanson math lectures introduces proof as the basis for establishing mathematical claims, rather than relying only on examples.
ChatGPT can generate proposed proofs, but its output can contain mistakes and needs checking.
Lean is described as software that checks proof steps, while emphasizing that human understanding remains important.

46:14 Every Nonzero Linear Equation Has a Real Solution

Unlike an equation such as x² = −1 over the real numbers, every nonzero linear equation in real variables has a solution.
With one variable, a₁x₁ = b and a₁ ≠ 0 has the unique solution x₁ = b/a₁.
With at least two variables, a nonzero linear equation has infinitely many solutions.

49:33 Solving for a Nonzero-Coefficient Variable

Choose an index j with aⱼ ≠ 0 in a₁x₁ + ⋯ + aₙxₙ = b.
Assign arbitrary real values to the other n − 1 variables, then solve for xⱼ by dividing by aⱼ.
The resulting formula is xⱼ = (b − Σⱼ≠ᵢ aᵢxᵢ)/aⱼ, which constructs solutions and explains why there are infinitely many when n ≥ 2.

56:30 From Linear Equations to Vector Algebra

The lecture moves from equation solution sets to vector spaces, vector subspaces, and affine subspaces in Rⁿ.
Vector addition, subtraction, and scalar multiplication are introduced as the basic operations on n-dimensional vectors.
These operations will help express solution spaces in more structured forms.

58:01 Coordinate-Wise Addition in R², R⁴, and Rⁿ

To add vectors, add corresponding coordinates: (3, −5) + (2, 11) = (5, 6).
In R⁴, (1, 2, 3, 4) + (5, 6, 7, 8) = (6, 8, 10, 12).
The same coordinate-wise rule applies to any pair of vectors in Rⁿ.

1:01:32 Coordinate-Wise Subtraction and the Column-Vector Rule

Subtract corresponding coordinates: (3, −5) − (2, 11) = (1, −16), and (1, 2, 3, 4) − (5, 6, 7, 8) = (−4, −4, −4, −4).
For the exercise (3, −2, 4) and (0, 5, 7), the sum is (3, 3, 11) and the difference is (3, −7, −3).
Vectors must be written as columns in this course because later matrix multiplication depends on that convention.

1:06:20 Vector Products Are Undefined; Scalar Multiplication Is Defined

The course does not define multiplication of one vector by another; multiplying matching coordinates is not the intended operation.
Scalar multiplication multiplies every coordinate by the same number: 7(3, −5) = (21, −35).
For example, (1, 2, 3, 4) scaled by 7 is (7, 14, 21, 28), while one-half of (3, −2, 4) is (3/2, −1, 2).

1:14:02 Writing Solution Spaces with Direction Vectors

The solutions of 4x + 7y = 0 can be written as all scalar multiples of (1, −4/7), or equivalently of (7, −4).
For 3x − 2y − 5z = 17, the solution space is (0, 0, −7/5) + x(1, 0, 3/5) + y(0, 1, −2/5), for real x and y.
This form separates one particular solution from all linear combinations of two direction vectors.

1:24:25 Next Class, Office Hours, Homework Submission, and Attendance

The next class will continue vector algebra and begin systems of linear equations; the example 3x + 2y = 18 and 4x + 3y = −5 previews the topic.
Office hours are Mondays and Wednesdays from 10:00 to 11:00, with additional one-on-one meetings available by email appointment.
Homework is submitted as one PDF through Brightpace; attendance is not graded, and missed lectures are available on YouTube.
Class is scheduled for 11:00–12:40 and will not run past 12:40.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Nathanson math lectures.