1 How to use this guide
Study one unit, solve its representative problem, and only then attempt the final quiz. The sequence follows the conceptual dependencies in the 3Blue1Brown Essence of Linear Algebra playlist.
2 Unit 1 — Vectors, combinations, and bases
A vector is an ordered list of scalars:
\[ \mathbf v=\begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix}. \]
The essential operations are
\[ \mathbf u+\mathbf v=\begin{bmatrix}u_1+v_1\\u_2+v_2\end{bmatrix}, \qquad c\mathbf v=\begin{bmatrix}cv_1\\cv_2\end{bmatrix}. \]
A linear combination has the form
\[ c_1\mathbf v_1+\cdots+c_k\mathbf v_k. \]
A basis is a linearly independent spanning set. Coordinates tell us which linear combination of the basis vectors produces a vector.
2.1 Representative problem
Find (a,b) such that
\[ a\begin{bmatrix}1\\2\end{bmatrix}+b\begin{bmatrix}2\\-1\end{bmatrix}=\begin{bmatrix}7\\1\end{bmatrix}. \]
3 Unit 2 — Linear transformations and matrices
A transformation (T) is linear when
\[ T(\mathbf u+\mathbf v)=T(\mathbf u)+T(\mathbf v), \qquad T(c\mathbf v)=cT(\mathbf v). \]
Every linear transformation can be written as
\[ T(\mathbf x)=A\mathbf x. \]
The columns of (A) are the images of the standard basis vectors. Matrix multiplication is composition:
\[ (AB)\mathbf x=A(B\mathbf x). \]
For a (2) matrix,
\[ A=\begin{bmatrix}a&b\\c&d\end{bmatrix}, \qquad \det(A)=ad-bc. \]
The matrix is invertible exactly when its determinant is nonzero.
3.1 Representative problem
For
\[ A=\begin{bmatrix}2&1\\0&3\end{bmatrix}, \]
compute (A[1,2]^T), (det(A)), and decide whether (A) is invertible.
4 Unit 3 — Systems, dot products, and cross products
A linear system is represented by
\[ A\mathbf x=\mathbf b. \]
The dot product is
\[ \mathbf u\cdot\mathbf v=\sum_i u_iv_i=\|\mathbf u\|\|\mathbf v\|\cos\theta. \]
Perpendicular vectors have dot product zero.
In three dimensions,
\[ \mathbf u\times\mathbf v= \begin{bmatrix} u_2v_3-u_3v_2\\ u_3v_1-u_1v_3\\ u_1v_2-u_2v_1 \end{bmatrix}. \]
The cross product is perpendicular to both inputs, and
\[ \|\mathbf u\times\mathbf v\|=\|\mathbf u\|\|\mathbf v\|\sin\theta. \]
4.1 Representative problem
For (mathbf u=[1,2,0]^T) and (mathbf v=[2,-1,1]^T), compute (mathbf uv) and (mathbf uv).
5 Unit 4 — Determinants, basis changes, and eigenvalues
Important determinant identities are
\[ \det(AB)=\det(A)\det(B), \qquad \det(A^{-1})=\frac1{\det(A)}. \]
Eigenvectors satisfy
\[ A\mathbf v=\lambda\mathbf v, \qquad \mathbf v\ne\mathbf0. \]
Eigenvalues solve the characteristic equation
\[ \det(A-\lambda I)=0. \]
If (A) has enough independent eigenvectors, then
\[ A=PDP^{-1}, \qquad A^n=PD^nP^{-1}. \]
For a (2) matrix,
\[ \lambda_1+\lambda_2=\operatorname{tr}(A), \qquad \lambda_1\lambda_2=\det(A). \]
5.1 Representative problem
Find the eigenvalues and eigenvectors of
\[ A=\begin{bmatrix}3&1\\0&2\end{bmatrix}. \]
6 Unit 5 — Abstract vector spaces
Vectors need not be arrows: polynomials, matrices, functions, and sequences can also form vector spaces. For the polynomial basis
\[ \mathcal B=\{1,x,x^2\}, \]
the polynomial (p(x)=4-2x+3x^2) has coordinate vector
\[ [p]_{\mathcal B}=\begin{bmatrix}4\\-2\\3\end{bmatrix}. \]
The same ideas—span, independence, basis, coordinates, and linear transformations—apply in every vector space.
6.1 Representative problem
Express (p(x)=2+5x-x^2) in coordinates relative to
\[ \mathcal B=\{1+x,x,x^2\}. \]
7 Quiz
Attempt these closed-notes first. Each question is worth 2 points.
- Find (a,b) such that (a[1,2]T+b[2,-1]T=[7,1]^T).
- State the two conditions a set must satisfy to be a basis.
- Find the matrix for (T(x,y)=(x+2y,3x-y)).
- Compute \[ \begin{bmatrix}1&2\\3&4\end{bmatrix}\begin{bmatrix}2\\1\end{bmatrix}. \]
- Solve (x+y=5) and (2x-y=1).
- Are ([1,2,3]^T) and ([2,-1,0]^T) perpendicular?
- Compute ([1,0,1]TT).
- Find the determinant of ([]).
- Find the eigenvalues of ([]).
- Express (2+5x-x^2) in the basis ({1+x,x,x^2}).
- Explain why matrix multiplication represents composition of transformations.
- Explain why the determinant being zero prevents invertibility.
7.1 Scoring
- 20–24: ready for a cumulative problem set.
- 14–19: review the missed units and retry.
- 0–13: work through the representative problems slowly, then retake the quiz.
The answer key gives worked solutions.