Linear Algebra - Lecture 11 - Fall, 2026

Overview

The lecture reviews vector spaces, systems of linear equations, Gaussian elimination, matrix operations, and linear transformations, then develops eigenvalues and the basis-and-dimension theorems. Worked examples include solving an 8x + 3y = 0, 5x + 2y = 1 system by both row reduction and a 2×2 inverse, and deriving the eigenvalues 5 ± 2√6; the class also reviews concepts and expectations for the midterm scheduled for the 21st.

Key takeaways

A matrix equation Ax = b can be solved by row-reducing the augmented matrix or, when A is square and invertible, by computing x = A⁻¹b; both methods give (−3, 8) for the worked system.
Homogeneous systems connect equation counting to vector dependence: more variables than equations guarantee a nonzero solution, a fact used to prove that bases of the same subspace have equal size.
A matrix A defines a linear transformation because matrix multiplication preserves both vector addition and scalar multiplication.
For A = [[8, 3], [5, 2]], nonzero eigenvectors require det(λI − A) = 0, producing the eigenvalues 5 ± 2√6.
The rotation by θ in the plane has matrix [[cos θ, −sin θ], [sin θ, cos θ]], whose columns are the rotated standard basis vectors.

Chapters

0:00 Vector Addition, Scalar Multiplication, and Vector-Space Axioms

In Rⁿ, vectors are n-coordinate columns; addition combines matching coordinates, and scalar multiplication multiplies every coordinate by the same real number.
Vector addition is associative and commutative, has a zero vector, and gives every vector an additive inverse.
Scalar multiplication interacts with vector addition and scalar products through the standard vector-space rules.

7:59 Vector Spaces Beyond Rⁿ: Functions, Polynomials, and Matrices

Continuous real-valued functions form a vector space because sums and scalar multiples of continuous functions remain continuous.
Polynomials are closed under addition and scalar multiplication; the examples 3x³ − 7x + 19 and x⁵ − x³ + 1 sum to x⁵ + 2x³ − 7x + 20.
The set of m×n matrices is a vector space, and Rⁿ can be viewed as the space of n×1 matrices.

12:58 Homogeneous Systems Have Subspaces of Solutions

A system with m equations and n variables is homogeneous when every right-hand side bᵢ equals zero.
The solution set of a homogeneous system is a subspace of Rⁿ.
A consistent inhomogeneous system has an affine solution set: a translate of a subspace; an inhomogeneous system may instead be inconsistent.

15:42 Equivalent Equations and the Three Gaussian-Elimination Operations

Interchanging two equations preserves the system's solutions.
Multiplying an equation by a nonzero scalar preserves its solutions.
Adding a scalar multiple of one equation to a different equation is the third solution-preserving operation used in Gaussian elimination.

22:03 Augmented Matrices and Row-Reduced Echelon Form

An m-equation, n-variable system can be written as Ax = b, with A containing the variable coefficients and b the right-hand-side column.
The augmented matrix appends b to A, giving m rows and n + 1 columns.
Gaussian elimination applies elementary row operations to seek row-reduced echelon form: leading 1s move rightward down the rows, with zeros above and below each leading 1.

25:14 Matrix Row Operations Mirror Equation Operations

Swapping equations corresponds to swapping rows of the augmented matrix.
Scaling an equation by a nonzero number corresponds to scaling its matrix row.
Adding a multiple of one equation to another corresponds to a row replacement, preserving the system's solution set.

28:27 Putting a 2×2 Matrix into Reduced Echelon Form

For the matrix with rows (3, 2) and (5, −1), scale the first row by 1/3 to create a leading 1.
Subtract five times the first row from the second, then scale the resulting second row to create its leading 1.
Clear the entry above the second pivot to obtain the identity matrix, illustrating the leading-one and zero-column conditions.

31:08 Solve 8x + 3y = 0 and 5x + 2y = 1 by Row Reduction

The augmented matrix is [[8, 3, 0], [5, 2, 1]].
Row operations reduce it to [[1, 0, −3], [0, 1, 8]], so the solution is x = −3 and y = 8.
Substitution verifies both equations: 8(−3) + 3(8) = 0 and 5(−3) + 2(8) = 1.

34:37 Solve the Same System Using a 2×2 Matrix Inverse

The coefficient matrix is [[8, 3], [5, 2]], with determinant 8·2 − 3·5 = 1.
Because the determinant is nonzero, the inverse exists; for a 2×2 matrix, swap the diagonal entries and negate the off-diagonal entries, then divide by the determinant.
Multiplying the inverse by the right-hand side vector (0, 1)ᵀ gives (−3, 8)ᵀ, matching row reduction.

40:12 Transpose and Symmetric Matrices

Transposition exchanges rows and columns: a 3×2 matrix becomes a 2×3 matrix.
A square matrix is symmetric when Aᵀ = A, equivalently when entries reflected across the main diagonal match.
The example [[1, 2, 3], [2, 4, 5], [3, 5, 6]] is symmetric.

43:21 Every Matrix Defines a Linear Transformation

An m×n matrix A maps a vector x in Rⁿ to Ax in Rᵐ, defining a function Tₐ: Rⁿ → Rᵐ.
A linear transformation preserves addition, T(x + y) = T(x) + T(y), and scalar multiplication, T(cx) = cT(x).
Matrix multiplication satisfies both rules, so every matrix-defined map is linear; finite-dimensional linear transformations can be represented by matrices.

48:20 Eigenvalues from the Equation Ax = λx

For A = [[8, 3], [5, 2]], an eigenvector x ≠ 0 must satisfy Ax = λx.
Rearranging gives (λI − A)x = 0; a nonzero solution exists only when det(λI − A) = 0.
The characteristic equation is λ² − 10λ + 1 = 0, yielding eigenvalues λ = 5 ± 2√6.

58:17 Plane Rotations as 2×2 Matrices

Rotating the unit vector (1, 0) by π/6 gives (√3/2, 1/2); rotating (0, 1) gives (−1/2, √3/2).
These images of the standard basis vectors form the columns of the rotation matrix [[cos θ, −sin θ], [sin θ, cos θ]].
The class is told that matrix representations of rotations will be developed further after the midterm.

1:04:27 Midterm Review: Linear Combinations and Independence

A linear combination of v₁,…,vₖ has the form c₁v₁ + ··· + cₖvₖ.
The vectors are linearly independent if the zero vector can be expressed as their linear combination only when every coefficient is zero.
They are dependent if some combination equals zero with at least one nonzero coefficient.

1:08:19 Subspaces, Span, and the Definition of a Basis

A subspace contains the zero vector and is closed under vector addition and scalar multiplication.
The span of a nonempty set S consists of all linear combinations of vectors in S; S generates W when its span equals W.
A basis for W both generates W and is linearly independent.

1:15:21 Why Dimension Is Well-Defined

A homogeneous system with n variables and m equations has a nonzero solution whenever n > m.
If W is generated by m vectors, any set of more than m vectors in W must be linearly dependent: express each vector using the generators and obtain m homogeneous equations in more than m coefficients.
For two bases of W with sizes m and n, each basis's independence and the other's spanning property imply m ≤ n and n ≤ m; therefore m = n, defining dimension consistently.

1:25:50 Midterm Format and Final Review Guidance

The midterm is scheduled for the 21st on campus, with review problems posted on Brightspace and emailed to students.
The exam is expected to have about eight required problems, potentially with extra choices; most will test computations such as solving systems or multiplying matrices.
Some elementary proofs will appear, but the stated choice structure allows students to omit a proof problem and still earn full credit.

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.