Linear AlgebraCollege3 topic explainers

Linear Algebra course

Matrix computation and the geometry of vectors.

Topics in this course

1.Vectors
A vector has both size and direction and is written in components such as $\mathbf{v} = \langle 3, 4 \rangle$ . Vectors add tip to tail, component by component, and scalar multiplication stretches or shrinks them. The magnitude $|\mathbf{v}| = \sqrt{3^2 + 4^2} = 5$ is the length.
- Unit vector: $\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}$ .
- Angle between vectors: $\cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}$ .
- Projection of $\mathbf{u}$ onto $\mathbf{v}$ : $\frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2}\mathbf{v}$ .
Quick check: Find $|\langle 6, 8 \rangle|$ .
$10$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study vectors
2.Matrices
A matrix is a rectangular array of numbers that represents a linear transformation or a system of equations. Matrices of the same size add entry by entry. Multiplying $AB$ takes dot products of rows of $A$ with columns of $B$ , so the number of columns of $A$ must equal the number of rows of $B$ , and in general $AB \ne BA$ .
- For $2 \times 2$ : $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$ .
- $2 \times 2$ inverse: $\frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ .
- Row reduction solves systems of any size and finds rank and inverses.
Quick check: Find $\det\begin{pmatrix} 4 & 3 \\ 2 & 5 \end{pmatrix}$ .
$14$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study matrices
3.Eigenvalues and Eigenvectors
An eigenvector of a square matrix $A$ is a nonzero vector that $A$ only stretches, without changing its direction: $A\mathbf{v} = \lambda\mathbf{v}$ . The scale factor $\lambda$ is the eigenvalue. Eigenvectors reveal the natural axes of a transformation, which is why they appear in vibration analysis, Google's PageRank and principal component analysis.
- Characteristic equation: $\det(A - \lambda I) = 0$ .
- Sum of eigenvalues = trace; product = determinant.
- Triangular matrices have their eigenvalues on the diagonal.
Quick check: Find the larger eigenvalue of $\begin{pmatrix} 3 & 0 \\ 0 & 7 \end{pmatrix}$ .
$7$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study eigenvalues and eigenvectors

Topics in this course

1.Vectors

2.Matrices

3.Eigenvalues and Eigenvectors

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