Linear Algebra - Lecture 3 - Fall 2026

Overview

Nathanson math lectures connects linear combinations and systems of equations to subspaces, showing that solutions to homogeneous systems form subspaces while consistent inhomogeneous systems have affine solution sets. The lecture develops Gaussian elimination through row operations and examples with two, three, and four variables, including how free variables describe families of solutions.

Key takeaways

A homogeneous linear system always has the zero solution, and its full solution set is a subspace because it is closed under addition and scalar multiplication.
A consistent inhomogeneous system has an affine solution set: one particular solution plus the subspace of solutions to the associated homogeneous system.
Gaussian elimination preserves the solution set through equation swaps, multiplication by nonzero scalars, and adding a multiple of one equation to another.
When a system has free variables, assigning parameters to them expresses all solutions as a particular vector plus linear combinations of direction vectors.
A row-reduced contradiction such as 0=1 proves inconsistency; a row with a free variable instead signals a family of solutions.

Chapters

0:00 Linear Combinations and Vector Coordinates

In Rⁿ, vectors are columns of n real coordinates, and scalars are real numbers.
A linear combination is formed by multiplying finitely many vectors by scalars and adding the results.
The definition is foundational and should be recalled without notes for exams.

5:00 Writing E₁ as a Combination of F₁ and F₂

For E₁=(1,0), E₂=(0,1), F₁=(1,1), and F₂=(1,2), the lecture first writes F₁=E₁+E₂ and F₂=E₁+2E₂.
To express E₁ using F₁ and F₂, set c₁F₁+c₂F₂=E₁ and match coordinates.
Solving c₁+c₂=1 and c₁+2c₂=0 gives c₁=2 and c₂=−1; checking yields 2F₁−F₂=(1,0).

12:00 Solving a Two-Equation System by Elimination

For 3x+2y=18 and 4x+3y=−5, multiplying by 3 and −2 respectively eliminates y.
The resulting solution is x=64 and y=−87, verified by substitution in both original equations.
A system is consistent if it has at least one solution and inconsistent if it has none.

18:00 Coefficient Notation and Homogeneous Systems

For m equations in n variables, aᵢⱼ denotes the coefficient of xⱼ in equation i, with right-hand side bᵢ.
A system is homogeneous when every bᵢ is zero; setting every variable to zero always gives a solution.
An inhomogeneous system has at least one nonzero right-hand side and may be inconsistent.

24:10 Subspaces: Closure and the Zero Vector

A subset W of a vector space V is a subspace when it contains the zero vector and is closed under vector addition and scalar multiplication.
Closure under those two operations is equivalent to closure under taking linear combinations.
These conditions distinguish subspaces from arbitrary subsets of Rⁿ.

29:00 Why Homogeneous Solution Sets Are Subspaces

Represent each solution as a column vector in Rⁿ and let W be the set of solutions to a homogeneous system.
The zero vector satisfies every equation because each left-hand side evaluates to zero.
If x and y solve the system, linearity gives A(x+y)=Ax+Ay=0; for any scalar c, A(cx)=cAx=0.
Therefore W contains zero and is closed under addition and scalar multiplication.

36:00 A Homogeneous System Producing a Line in R³

For x+2y+4z=0 and x+3y+9z=0, subtracting the equations gives y=−5z.
Substitution gives x=6z, so every solution has the form z(6,−5,1).
The solution set is a one-dimensional subspace: a line through the origin in R³.

43:00 Geometric Examples of Homogeneous Solution Spaces

The vector (6,−5,1) describes the direction of the solution line through the origin in three-dimensional space.
For the single equation 7x+2y=0, every solution is x(1,−7/2).
That two-variable homogeneous system also has a line through the origin as its solution space.

48:00 Translating a Subspace Creates an Affine Subspace

For a vector v and subspace W, the translate v+W is the set of all vectors v+w with w in W.
In R², translating the line of multiples of (1,2) by v=(3,0) shifts it three units to the right.
The translated line is an affine subspace; unlike the original subspace, it generally does not pass through the origin.

53:00 Equivalent Systems and the Goal of Gaussian Elimination

Two systems in the same variables are equivalent when they have exactly the same solutions, even if they contain different numbers of equations.
Gaussian elimination repeatedly replaces a system with a simpler equivalent system until its solutions can be read off.
For a consistent inhomogeneous system, the solution set is an affine subspace.

58:00 Three Row Operations Preserve Solutions

Interchanging two equations preserves the solution set because it only changes their order.
Multiplying an equation by a nonzero scalar preserves its solutions; for example, x+7y=67 can be multiplied by 3.
Replacement adds a scalar multiple of one equation to a different equation, as when −2 times one row is added to another.
Using these operations, the example system reduces to y=9 and x=4, so its solution is (4,9).

1:03:00 A Worked Row-Replacement Example

In the three-equation example, swapping equations and multiplying by 3 creates a convenient first row, 3x+21y=201.
Adding −2 times the second equation to the third eliminates x and yields −27y=−243.
Back-substitution gives y=9 and then x=4, illustrating how equivalent systems expose the solution.

1:07:00 Gaussian Elimination with a Two-by-Two System

For 9x+11y=−1 and 5x+6y=−3, scaling the first equation creates a leading coefficient of 1 for x.
Subtracting five times the scaled first equation from the second eliminates x, demonstrating the pivot-and-eliminate strategy.
The lecture emphasizes exact fraction arithmetic and checking the final values in the original equations.

1:14:00 Using a Free Variable in a Two-Equation, Three-Unknown System

The example system is 2x+3y+z=1 and 5x+y−4z=−1, with three variables but only two equations.
After elimination, the equations can be written as y+z=7/13 and x+3y+z=1.
Taking z=t as a free parameter gives (x,y,z)=(-4/13,7/13,0)+t(1,-1,1).
The solution set is an affine line: one particular solution plus all multiples of a homogeneous direction vector.

1:19:00 Reading the Three-Variable Solution as an Affine Line

The parameter t can take any real value, so the two-equation system has infinitely many solutions.
The vector (-4/13,7/13,0) is a particular solution, and (1,-1,1) gives the direction of the solution line.
Gaussian elimination exposes both the free variable and the affine structure of the solution set.

1:23:30 Solving Four Variables with Two Independent Parameters

For x+y+z+w=10 and x−y+z−w=−2, elimination yields x+z=4 and y+w=6.
Choosing z and w freely gives (x,y,z,w)=(4,6,0,0)+z(-1,0,1,0)+w(0,-1,0,1).
The solutions form an affine plane: a particular vector plus linear combinations of two direction vectors.

1:29:00 Gaussian Elimination Practice and Lecture Wrap-Up

Gaussian elimination scales to systems with multiple variables and identifies free variables when there are fewer independent equations.
A contradiction such as 0=1 signals that the system is inconsistent and has no solutions.
The assigned practice focuses on Gaussian elimination and systems of equations, with a reminder that repeated work builds arithmetic fluency.

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.