Linear Algebra Interactive Learning Hub

Module 1

Vectors, Linear Combinations, Span & Independence

Understand how vectors add, scale, and span 2D space, and visualize when vectors become linearly dependent.

Vectors & Linear Combinations

Vector (向量): /ˈvektər/ An object with magnitude and direction, represented as an arrow in space.
$$\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2$$
Linear Combination (线性组合): /ˈlɪniər ˌkɒmbɪˈneɪʃn/ Sum of scalar multiples of vectors: $c_1 \mathbf{u} + c_2 \mathbf{v}$.

Span & Linear Independence

Span (张成空间): /spæn/ The set of all possible linear combinations formed by a set of vectors.
$$\text{Span}(\mathbf{u}, \mathbf{v}) = \{ c_1 \mathbf{u} + c_2 \mathbf{v} \mid c_1, c_2 \in \mathbb{R} \}$$
Linear Independence (线性无关): /ˈlɪniər ˌɪndɪˈpendəns/ No vector in the set can be written as a linear combination of the others ($c_1 \mathbf{u} + c_2 \mathbf{v} = \mathbf{0} \implies c_1 = c_2 = 0$).

Subspace, Basis & Dimension

Subspace (子空间): /ˈsʌbˌspeɪs/ A subset $W \subseteq V$ containing $\mathbf{0}$, closed under vector addition and scalar multiplication.
Basis & Dimension (基与维度): /ˈbeɪsɪs/ /daɪˈmenʃn/ A basis is a linearly independent set that spans the space. Dimension is the number of basis vectors.
$$\dim(\mathbb{R}^2) = 2, \quad \mathbf{e}_1 = \begin{bmatrix}1\\0\end{bmatrix}, \mathbf{e}_2 = \begin{bmatrix}0\\1\end{bmatrix}$$

Vector u: ($u_x$, $u_y$)

$u_x$: 2.0

$u_y$: 1.0

Vector v: ($v_x$, $v_y$)

$v_x$: -1.0

$v_y$: 2.0

Multipliers ($c_1, c_2$):

$c_1$: 1.0

$c_2$: 1.0

$c_1 \mathbf{u} + c_2 \mathbf{v} =$: [1.00, 3.00]

$\det([\mathbf{u}\ \mathbf{v}]) =$: 5.00

Vectors are Linearly Independent: $\det \neq 0$, Span is the full 2D plane ($\mathbb{R}^2$).

💡 Drag vector tips on canvas

Module 2

Matrices as Linear Transformations & Determinants

Visualize how a 2x2 matrix moves, skews, and rotates space, and discover why Determinant equals area scaling factor.

Linear Transformations

Linear Transformation (线性变换): /ˈlɪniər ˌtrænsfərˈmeɪʃn/ A mapping that preserves grid lines (lines remain lines, origin remains fixed).
$$T(\mathbf{x}) = A\mathbf{x} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$$
Columns of Matrix = Transformed Basis Vectors: Column 1 is $T(\hat{i})$, Column 2 is $T(\hat{j})$.

Determinant, Rank & Nullspace

Determinant (行列式): /dɪˈtɜːrmɪnənt/ The factor by which a transformation scales areas ($\det(A) = ad - bc$).
$$\det(A) = ad - bc$$
Rank & Nullspace (秩与零空间): /ræŋk/ /ˈnʌlspeɪs/ Rank is the dimension of output space; Nullspace is all vectors that land on origin ($\mathbf{0}$).

Composition & Inverse Matrix

Matrix Composition (矩阵乘法与复合变换): Multiplying $BA$ applies transformation $A$ first, then $B$. Note $BA \neq AB$ (non-commutative).
Inverse Matrix (逆矩阵): /ˈɪnvɜːrs/ Undoes the transformation. Exists iff $\det(A) \neq 0$:
$$A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

Matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$:

Preset Transformations:

$\det(A) = ad - bc =$: 1.80

$\text{Trace}(A) = a + d =$: 2.70

$\text{Rank}(A) =$: 2

Transformation is orientation-preserving: $\det(A) > 0$. Area is scaled by 1.80x.

First matrix A:

Second matrix B:

Application order:

Preset pairs:

Current product: BA = [[1, 1], [-1, 0]]

Read from right to left: A acts first, then B.

Row × Column Derivation

Each result entry is one row of the left matrix dotted with one column of the right matrix.

BA = B × A

Matrix A:

Preset transformations:

Determinant: det(A) = 1

Apply A, then A⁻¹; the composition returns to the identity I.

Inverse Formula & Verification

A⁻¹ = 1 / det(A) · [[d, −b], [−c, a]]

Adjugate part[[d, −b], [−c, a]] = [[0, 1], [−1, 0]]

Inverse matrixA⁻¹ = [[0, 1], [−1, 0]]

AA⁻¹ = [[1, 0], [0, 1]]; A⁻¹A = [[1, 0], [0, 1]]

The inverse returns the unit square to its original position.

Module 3

Linear Systems, Rank & Gaussian Elimination

Use 2D lines, augmented matrices, and row operations to distinguish unique, no, and infinitely many solutions.

Geometry of Linear Systems

Each two-variable linear equation is a line; their intersection satisfies both equations.

Gauss-Jordan Elimination

Use row swaps, scaling, and replacement to reduce an augmented matrix to RREF.

Rank & Solution Classification

Different ranks for A and [A|b] mean no solution; equal ranks give a unique solution only when they match the number of unknowns.

Coefficient Matrix A and Constant Vector b:

Equations:

Current Augmented Matrix:

Current Row Operation:

rank(A):2

rank([A|b]):2

Module 4

Eigenvalues, Eigenvectors & Diagonalization

Discover vectors that do not change direction during transformation ($A\mathbf{v} = \lambda \mathbf{v}$) and explore spectral decomposition.

Eigenvalues & Eigenvectors

Eigenvalue & Eigenvector (特征值与特征向量): /ˈaɪɡənˌvæljuː/ /ˈaɪɡənˌvektər/ A non-zero vector $\mathbf{v}$ is an eigenvector if $A\mathbf{v}$ is a scalar multiple of $\mathbf{v}$.
$$A\mathbf{v} = \lambda \mathbf{v} \iff (A - \lambda I)\mathbf{v} = \mathbf{0}$$
Characteristic Equation (特征方程): $\det(A - \lambda I) = 0 \implies \lambda^2 - \text{tr}(A)\lambda + \det(A) = 0$.

Diagonalization & Spectral Theorem

Diagonalization (对角化): /daɪˈæɡənəlaɪˌzeɪʃn/ Factoring matrix $A$ into eigenvector basis $P$ and diagonal matrix $D$: $A = PDP^{-1}$.
$$A = P D P^{-1}, \quad A^n = P D^n P^{-1}$$
Spectral Theorem (谱定理): Every real symmetric matrix ($A = A^T$) can be orthogonally diagonalized ($A = Q D Q^T$) with real eigenvalues.

Multiplicity & PCA Connection

Algebraic vs Geometric Multiplicity: Algebraic is root multiplicity in characteristic eq; Geometric is dimension of eigenspace $\text{Null}(A-\lambda I)$.
PCA & Covariance Matrix (主成分分析): Principal components are eigenvectors of covariance matrix $C = \frac{1}{n} X^T X$, pointing in directions of max variance.

Symmetric Matrix $A = \begin{bmatrix} a & b \\ b & c \end{bmatrix}$:

Probe Vector Angle ($\theta$):

32°

Eigenvalue $\lambda_1 =$: 3.62

Eigenvalue $\lambda_2 =$: 1.38

Collinearity Angle Error: 14.2°

Probe $\mathbf{x}$ is not aligned with $A\mathbf{x}$. Keep rotating to align!

Module 5

Orthogonality, Projection, QR, PCA & SVD

Master the geometry of orthogonal projection, QR, least squares, PCA dimensionality reduction, and Singular Value Decomposition.

Dot Product & Projection

Dot Product & Orthogonality (点积与正交性): /ɔːrˌθɒɡəˈnæləti/ Vectors $\mathbf{u}, \mathbf{v}$ are orthogonal ($\mathbf{u} \perp \mathbf{v}$) iff $\mathbf{u} \cdot \mathbf{v} = 0$.
$$\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos\theta$$
Orthogonal Projection (正交投影): /prəˈdʒekʃn/ The shadow of vector $\mathbf{v}$ onto line spanned by $\mathbf{u}$.
$$\text{proj}_{\mathbf{u}}\mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\|\mathbf{u}\|^2} \mathbf{u}$$

Gram-Schmidt, QR & SVD

Gram-Schmidt & QR (正交化与 QR 分解): /ɡræm ʃmɪt/ Converting vectors into orthonormal basis $Q$. Matrix form: $A = QR$.
Singular Value Decomposition (SVD 奇异值分解): /ˈsɪŋɡjələr ˈvæljuː/ Factoring any matrix $A = U \Sigma V^T$ into Rotation ($V^T$) $\to$ Scaling ($\Sigma$) $\to$ Rotation ($U$).

Least Squares & Pseudo-Inverse

Least Squares Approximation (最小二乘法): Solving inconsistent $A\mathbf{x} = \mathbf{b}$ by projecting $\mathbf{b}$ onto $\text{Col}(A)$:
$$A^T A \hat{\mathbf{x}} = A^T \mathbf{b} \implies \hat{\mathbf{x}} = (A^T A)^{-1} A^T \mathbf{b}$$
Moore-Penrose Pseudo-Inverse (伪逆): $A^+ = V \Sigma^+ U^T$, generalizes inverse to non-square matrices.

Base Vector u: ($u_x, u_y$)

$u_x$: 3.0

$u_y$: 0.0

Target Vector v: ($v_x, v_y$)

$v_x$: 2.0

$v_y$: 3.0

$\text{proj}_{\mathbf{u}}\mathbf{v} =$: [2.00, 0.00]

$\mathbf{v}^{\perp} = \mathbf{v} - \text{proj} =$: [0.00, 3.00]

Dot Product $\mathbf{u} \cdot \mathbf{v} =$: 6.00

Orthogonality Check: $(\text{proj}_{\mathbf{u}}\mathbf{v}) \cdot \mathbf{v}^{\perp} = 0.00$ (Perfect Right Angle!).

💡 Drag vector tips on canvas

First column vector u: ($u_x, u_y$)

$u_x$:3.0

$u_y$:1.0

Second column vector v: ($v_x, v_y$)

$v_x$:1.0

$v_y$:3.0

‖u‖ =3.162

r₁₂ = q₁ · v =1.897

‖u₂‖ =2.530

QᵀQ:[[1, 0], [0, 1]] Upper-triangular R:[[3.162, 1.897], [0, 2.530]] Reconstructed QR:[[3, 1], [1, 3]]

Step 0: use u and v as the two columns of A.

💡 Drag either source vector to restart the QR steps

Fitted Model:ŷ = 1.1x + 0.6

Residual Sum of Squares (SSE):0.12

Coefficient of Determination (R²):0.99

Normal Equation AᵀAθ = Aᵀy:

💡 Drag any point to refit the line

Mean μ:[0, 0]

Eigenvalues:λ₁ = 0, λ₂ = 0

Explained Variance (PC₁ / PC₂):—

Covariance Matrix C:[[0, 0], [0, 0]] First Principal Direction:[1, 0]

💡 Drag any point to recompute PCA

Matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$:

Decomposition Step View:

Singular Value $\sigma_1 =$: 2.04

Singular Value $\sigma_2 =$: 0.94

Condition Num $\kappa(A) = \sigma_1/\sigma_2$: 2.17

Right singular vectors $v_1, v_2$ map to orthogonal principal axes of output ellipse $\sigma_1 u_1, \sigma_2 u_2$.

Module 6

Linear Algebra Quiz & Formula Cheatsheet

Test your linear algebra knowledge with 15 practice questions or review key formulas, SVD intuitions, and common pitfalls.

Question 1 of 15 Vectors & Span

Score: 0 / 100

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Matrix Operations & Properties

Determinant Properties:

$$\det(AB) = \det(A)\det(B), \quad \det(A^T) = \det(A), \quad \det(cA) = c^n \det(A)$$

Trace Properties:

$$\text{tr}(AB) = \text{tr}(BA), \quad \text{tr}(A) = \sum_{i=1}^n \lambda_i$$

Rank-Nullity Theorem: /ræŋk ˈnʌləti/

$$\text{Rank}(A) + \text{Nullity}(A) = n \quad (\text{for } m \times n \text{ matrix } A)$$

Four Fundamental Subspaces

Strang's Fundamental Subspaces Theorem:

1. Column Space $\text{Col}(A)$: Dim = $r$, subspace of $\mathbb{R}^m$.
2. Nullspace $\text{Null}(A)$: Dim = $n-r$, subspace of $\mathbb{R}^n$. $\text{Null}(A) \perp \text{Row}(A)$.
3. Row Space $\text{Row}(A) = \text{Col}(A^T)$: Dim = $r$, subspace of $\mathbb{R}^n$.
4. Left Nullspace $\text{Null}(A^T)$: Dim = $m-r$, subspace of $\mathbb{R}^m$. $\text{Null}(A^T) \perp \text{Col}(A)$.

Positive Definite Matrix

Positive Definite /ˈpɒzətɪv ˈdefɪnət/

A real symmetric matrix $A$ is positive definite ($A \succ 0$) iff any of the following equivalent conditions holds:

1. $\mathbf{x}^T A \mathbf{x} > 0$ for every non-zero vector $\mathbf{x} \neq \mathbf{0}$.
2. All eigenvalues $\lambda_i > 0$ are strictly positive.
3. All leading principal minors are strictly positive ($\det(A_k) > 0$).
4. A Cholesky factorization exists: $A = L L^T$.

Singular Value Decomposition (SVD)

Singular Value Decomposition /ˈsɪŋɡjələr ˈvæljuː/

For any $m \times n$ matrix $A$:

$$A = U \Sigma V^T$$

$U$: $m \times m$ orthogonal matrix of left singular vectors (eigenvectors of $A A^T$).
$\Sigma$: $m \times n$ diagonal matrix of singular values $\sigma_i = \sqrt{\lambda_i(A^T A)}$.
$V$: $n \times n$ orthogonal matrix of right singular vectors (eigenvectors of $A^T A$).

Least Squares & PCA Applications

Normal Equations:

$$A^T A \hat{\mathbf{x}} = A^T \mathbf{b}$$

Projection matrix onto $\text{Col}(A)$: $P = A(A^T A)^{-1}A^T$.

PCA (Principal Component Analysis):

Sample covariance $C = \frac{1}{n} X^T X$. Applying SVD to the centered data matrix $X = U \Sigma V^T$ gives right singular vectors $V$ as the principal directions.

Common Linear Algebra Pitfalls

Pitfall 1: Matrix Multiplication is Non-Commutative!

$AB \neq BA$ in general. $(AB)^T = B^T A^T$ and $(AB)^{-1} = B^{-1}A^{-1}$.

Pitfall 2: Eigenvalue vs. Singular Value—what is the difference?

Eigenvalues apply only to square matrices and can be complex or negative. Singular values apply to any rectangular matrix and are always non-negative real numbers $\sigma_i \ge 0$.

Understand Linear Algebra Through Geometry, Relationships, and Change