1 Scope
This guide follows the public lessons retrieved from the Linear Algebra playlist. Members-only and unavailable lessons are excluded.
2 Unit 1 — Foundations of vector spaces
Videos: 1, 2, 4, 5.
Vectors in (R^n) are tuples of scalars. A line through a point (p) with direction (v) is
\[ L=\{\mathbf p+t\mathbf v:t\in\mathbb R\}. \]
A vector space is closed under vector addition and scalar multiplication and satisfies the vector-space axioms. A subspace (UV) must contain (0) and satisfy
\[ \mathbf u,\mathbf v\in U,\ a,b\in\mathbb R \quad\Longrightarrow\quad a\mathbf u+b\mathbf v\in U. \]
Representative problem: determine whether the solution set of (x+2y-z=0) is a subspace of (R^3), and parameterize it.
3 Unit 2 — Subspaces, inner products, and geometry
Videos: 6, 9, 10.
The Euclidean inner product and norm are
\[ \langle\mathbf u,\mathbf v\rangle=\sum_i u_iv_i, \qquad \|\mathbf v\|=\sqrt{\langle\mathbf v,\mathbf v\rangle}. \]
The angle relation is
\[ \langle\mathbf u,\mathbf v\rangle=\|\mathbf u\|\|\mathbf v\|\cos\theta. \]
In (R^3), (uv) is perpendicular to both inputs and has magnitude equal to the parallelogram area:
\[ \|\mathbf u\times\mathbf v\|=\|\mathbf u\|\|\mathbf v\|\sin\theta. \]
Representative problem: compute the projection of (u) onto nonzero (v):
\[ \operatorname{proj}_{\mathbf v}(\mathbf u)= \frac{\langle\mathbf u,\mathbf v\rangle}{\langle\mathbf v,\mathbf v\rangle}\mathbf v. \]
4 Unit 3 — Matrices and linear maps
Videos: 11, 13, 14, 16, 17, 19, 20.
A linear map satisfies
\[ T(a\mathbf u+b\mathbf v)=aT(\mathbf u)+bT(\mathbf v). \]
For a matrix (A=[a_1 a_n]),
\[ A\mathbf x=x_1\mathbf a_1+\cdots+x_n\mathbf a_n. \]
Composition is matrix multiplication:
\[ (AB)\mathbf x=A(B\mathbf x). \]
The matrix of a linear map is determined by the images of the standard basis vectors. Special matrices include diagonal, triangular, symmetric, and identity matrices.
Representative problem: find the matrix of (T(x,y)=(2x-y,x+3y)), then compute (T(1,2)).
5 Unit 4 — Independence, bases, coordinates, and dimension
Videos: 22, 23, 25, 26, 28.
Vectors (v_1,,v_k) are linearly independent when
\[ c_1\mathbf v_1+\cdots+c_k\mathbf v_k=0 \quad\Longrightarrow\quad c_1=\cdots=c_k=0. \]
A basis is an independent spanning set. Coordinates relative to a basis (B=(b_1,,b_n)) satisfy
\[ \mathbf v=[\mathbf v]_B{}_1\mathbf b_1+\cdots+[\mathbf v]_B{}_n\mathbf b_n. \]
The exchange lemma explains why all bases of a finite-dimensional space have the same number of vectors: the dimension.
Representative problem: express ((5,1)) in the basis ({(1,1),(2,-1)}).
6 Unit 5 — Inverses and transposes
Videos: 29, 31, 32.
An inverse satisfies
\[ AA^{-1}=A^{-1}A=I. \]
For a (2) matrix,
\[ \begin{bmatrix}a&b\\c&d\end{bmatrix}^{-1} =\frac1{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}, \qquad ad-bc\ne0. \]
The transpose is defined by
\[ (A^T)_{ij}=A_{ji}, \qquad (AB)^T=B^TA^T. \]
Representative problem: find the inverse of ( \[\begin{bmatrix}2&1\\1&1\end{bmatrix}\]) and verify it by multiplication.
7 Unit 6 — Range, kernel, rank-nullity, and solving systems
Videos: 34, 35, 37, 38, 40, 42.
The kernel and range are
\[ \ker(A)=\{\mathbf x:A\mathbf x=0\}, \qquad \operatorname{range}(A)=\{A\mathbf x:\mathbf x\in V\}. \]
The rank-nullity theorem is
\[ \dim(V)=\operatorname{rank}(A)+\operatorname{nullity}(A). \]
Row operations preserve the solution set of a system when applied to its augmented matrix. In row-echelon form, pivot variables are determined and free variables parameterize the remaining solutions.
Representative problem: row-reduce an augmented matrix and state whether the system has no solution, one solution, or infinitely many solutions.
8 Unit 7 — Determinants
Videos: 43, 44, 46, 47, 48, 50, 51.
For a (2) matrix,
\[ \det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc. \]
The determinant is multilinear and alternating, and satisfies
\[ \det(AB)=\det(A)\det(B). \]
For an (nn) matrix, Laplace expansion along row (i) is
\[ \det(A)=\sum_j(-1)^{i+j}a_{ij}\det(A_{ij}). \]
Geometrically, (|(A)|) is volume scaling. A square matrix is invertible exactly when its determinant is nonzero.
Representative problem: compute the determinant of ( \[\begin{bmatrix}1&2&3\\0&4&5\\1&0&6\end{bmatrix}\]) by expansion or elimination.
9 Unit 8 — Eigenvalues and eigenvectors
Videos: 53, 54, 55.
An eigenvector is a nonzero vector satisfying
\[ A\mathbf v=\lambda\mathbf v. \]
Eigenvalues solve the characteristic equation
\[ p_A(\lambda)=\det(A-\lambda I)=0. \]
The algebraic multiplicity of an eigenvalue is its multiplicity as a root of (p_A). Eigenvectors for () form the eigenspace
\[ E_\lambda=\ker(A-\lambda I). \]
Representative problem: find the eigenvalues and eigenspaces of ( \[\begin{bmatrix}3&1\\0&2\end{bmatrix}\]).
10 Quiz
Attempt this closed-notes. Each question is worth 2 points.
- Parameterize the line through ((1,2)) parallel to ((3,-1)).
- Prove that the solution set of a homogeneous linear system is a subspace.
- Compute ((1,2,-1),(2,0,3)) and both vector norms.
- Compute ((1,2,0)(2,-1,1)).
- Find the matrix of (T(x,y)=(2x-y,x+3y)).
- Explain why the columns of a matrix determine its matrix-vector product.
- Test whether ((1,2)) and ((2,4)) are linearly independent.
- Find the coordinates of ((5,1)) in the basis ({(1,1),(2,-1)}).
- Find the inverse of ( \[\begin{bmatrix}2&1\\1&1\end{bmatrix}\] ).
- State and interpret the rank-nullity theorem.
- Describe how free variables appear in a row-echelon solution.
- Compute ( \[\begin{bmatrix}4&1\\2&3\end{bmatrix}\] ).
- State the determinant criterion for invertibility.
- Find the characteristic polynomial of ( \[\begin{bmatrix}3&1\\0&2\end{bmatrix}\] ).
- Find the eigenspaces of that matrix.
- Explain why eigenvectors belonging to distinct eigenvalues are linearly independent.
10.1 Scoring
- 28–32: ready for proof-heavy linear algebra.
- 20–27: review the units containing missed questions.
- Below 20: revisit Units 1, 3, 4, and 6 before continuing.
See the answer key after attempting the quiz.