Applied Linear Algebra for AI and Machine Learning FAQ

This FAQ addresses common questions about the course content, concepts, and applications. Questions are organized by category to help you find answers quickly.

Getting Started Questions

What is this course about?

This course provides a comprehensive introduction to linear algebra with a strong emphasis on practical applications in artificial intelligence, machine learning, and computer vision. You'll develop both theoretical understanding and hands-on skills through interactive microsimulations that bring abstract mathematical concepts to life. The course covers vectors, matrices, linear transformations, eigenvalues, matrix decompositions, and their applications in neural networks, generative AI, image processing, and autonomous systems.

Who is this course designed for?

This course is designed for:

Computer Science majors pursuing AI/ML specializations
Data Science students seeking mathematical foundations
Engineering students interested in robotics and autonomous systems
Applied Mathematics students wanting practical applications
Graduate students needing linear algebra foundations for research

The material is presented at a college undergraduate level, making it accessible to anyone with the prerequisites who wants to understand the mathematics behind modern AI systems.

What are the prerequisites for this course?

To succeed in this course, you should have:

College Algebra or equivalent: Familiarity with functions, equations, and basic mathematical notation
Basic programming experience: Python is recommended but not required
Familiarity with calculus concepts: Understanding of derivatives and integrals at a conceptual level

You don't need prior exposure to linear algebra—this course starts from the fundamentals and builds up systematically.

How is the course structured?

The course is divided into four major parts spanning 15 chapters:

Part 1: Foundations of Linear Algebra (Chapters 1-4): Vectors, matrices, systems of equations, and linear transformations
Part 2: Advanced Matrix Theory (Chapters 5-8): Determinants, eigenvalues, matrix decompositions, and abstract vector spaces
Part 3: Linear Algebra in Machine Learning (Chapters 9-12): ML foundations, neural networks, generative AI, and optimization
Part 4: Computer Vision and Autonomous Systems (Chapters 13-15): Image processing, 3D geometry, and sensor fusion

Each chapter includes interactive MicroSims to reinforce concepts through hands-on exploration.

How do I use the interactive MicroSims?

MicroSims are browser-based interactive simulations that let you visualize and experiment with linear algebra concepts. To use them:

Navigate to the MicroSims section in the sidebar
Select a simulation relevant to what you're studying
Use the sliders, buttons, and controls to adjust parameters
Observe how changes affect the visualization in real-time
Connect the visual behavior to the mathematical concepts you're learning

No software installation is required—all MicroSims run directly in your web browser.

Why is linear algebra important for AI and machine learning?

Linear algebra is the mathematical language in which modern AI systems are written. Understanding it enables you to:

Debug ML models by understanding what's happening mathematically inside them
Optimize performance by choosing efficient matrix operations and representations
Innovate by seeing new ways to apply linear algebra concepts to novel problems
Communicate with researchers and engineers using shared mathematical vocabulary
Adapt to new techniques that build on these foundational concepts

From the matrix multiplications in neural networks to the transformations in computer vision, virtually every AI algorithm relies heavily on linear algebra operations.

How long does it take to complete each chapter?

Each chapter is designed for approximately one week of study, including:

Reading the chapter content (2-3 hours)
Working through examples and exercises (2-3 hours)
Exploring interactive MicroSims (1-2 hours)
Completing practice problems (2-3 hours)

The entire course spans 15 weeks at this pace, though self-study learners can adjust their schedule as needed.

What software do I need?

For reading the textbook and using MicroSims, you only need a modern web browser. For hands-on programming exercises, you'll benefit from:

Python 3.x with the following libraries:
NumPy: For numerical computations and array operations
Matplotlib: For creating visualizations
scikit-learn: For machine learning examples
Optional: GPU access for deep learning exercises in later chapters

All code examples in the textbook use Python with NumPy.

How can I check my understanding of the material?

Each chapter provides multiple ways to assess your understanding:

Concept check questions embedded throughout the text
Interactive MicroSims where you can test your predictions
Practice problems with varying difficulty levels
The glossary for reviewing terminology
Quiz questions for self-assessment

Working through these resources actively, rather than passively reading, is the key to building deep understanding.

Where can I get help if I'm stuck?

If you're struggling with a concept:

Review the relevant glossary definitions for terminology clarity
Use the MicroSims to build geometric intuition
Re-read prerequisite material if foundational concepts are unclear
Check the learning graph to ensure you've covered prerequisite concepts
For textbook issues, report problems on the GitHub Issues page

Core Concept Questions

What is the difference between a scalar and a vector?

A scalar is a single numerical value representing magnitude only (like temperature or mass). A vector is an ordered collection of scalars that represents both magnitude and direction. While the scalar 5 tells you "how much," the vector (3, 4) tells you "how much and in which direction."

Example: Speed of 60 mph is a scalar; velocity of 60 mph heading northeast is represented as a vector with components in the x and y directions.

What is a matrix and how does it relate to vectors?

A matrix is a rectangular array of numbers arranged in rows and columns. You can think of a matrix as:

A collection of column vectors side by side
A collection of row vectors stacked vertically
A representation of a linear transformation

A matrix with dimensions m×n has m rows and n columns. When you multiply a matrix by a vector, you're applying a linear transformation that maps the input vector to an output vector.

Example: A 3×2 matrix contains 3 rows and 2 columns, and can be viewed as 2 column vectors in 3-dimensional space.

What does it mean for vectors to be linearly independent?

Vectors are linearly independent if no vector in the set can be written as a linear combination of the others. Equivalently, the only way to combine them to get the zero vector is with all zero coefficients.

Example: The vectors (1, 0) and (0, 1) are linearly independent because neither is a multiple of the other. However, (1, 2) and (2, 4) are linearly dependent because (2, 4) = 2·(1, 2).

Linear independence is crucial because independent vectors provide "new directions" in space, while dependent vectors are redundant.

What is a basis and why is it important?

A basis is a set of linearly independent vectors that span an entire vector space. Every vector in the space can be written as a unique linear combination of basis vectors. The number of vectors in a basis equals the dimension of the space.

The basis is important because it provides a coordinate system for the vector space. The standard basis in 3D consists of the unit vectors along each axis: (1,0,0), (0,1,0), and (0,0,1).

Example: Any point in 3D space can be written as x(1,0,0) + y(0,1,0) + z(0,0,1) where (x, y, z) are the coordinates.

What is the dot product and what does it tell us?

The dot product (also called inner product) of two vectors produces a scalar value computed as the sum of products of corresponding components:

\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n\]

Geometrically, the dot product equals |a||b|cos(θ) where θ is the angle between the vectors. This tells us:

If the dot product is positive, vectors point in similar directions
If the dot product is zero, vectors are perpendicular (orthogonal)
If the dot product is negative, vectors point in opposite directions

Example: The dot product of (1, 2) and (3, 4) is 1×3 + 2×4 = 11.

What is a linear transformation?

A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. If T is a linear transformation, then:

T(u + v) = T(u) + T(v) for all vectors u and v
T(cv) = cT(v) for all vectors v and scalars c

Every linear transformation can be represented by a matrix. Common examples include rotations, reflections, scaling, shearing, and projections.

Example: Rotating a 2D vector by 45° is a linear transformation represented by the rotation matrix [[cos(45°), -sin(45°)], [sin(45°), cos(45°)]].

What is the determinant and what does it tell us?

The determinant is a scalar value computed from a square matrix that tells us:

Whether the matrix is invertible (nonzero determinant = invertible)
The volume scaling factor of the associated transformation
The orientation change (negative determinant = orientation flip)

For a 2×2 matrix [[a, b], [c, d]], the determinant is ad - bc.

Example: A rotation matrix always has determinant 1 (preserves area and orientation). A reflection matrix has determinant -1 (preserves area but flips orientation).

What are eigenvalues and eigenvectors?

An eigenvector of a matrix A is a nonzero vector v that, when transformed by A, points in the same direction (or exactly opposite)—it only gets scaled by a factor λ called the eigenvalue:

\[A\mathbf{v} = \lambda\mathbf{v}\]

Eigenvectors reveal the "natural directions" of a transformation where the transformation acts as simple scaling rather than rotation or shearing.

Example: For a horizontal stretch matrix that doubles the x-coordinate, any vector along the x-axis is an eigenvector with eigenvalue 2, and any vector along the y-axis is an eigenvector with eigenvalue 1.

What is Singular Value Decomposition (SVD)?

SVD decomposes any matrix A into three matrices: A = UΣV^T where:

U contains left singular vectors (orthonormal columns)
Σ is a diagonal matrix of singular values (non-negative, decreasing)
V^T contains right singular vectors (orthonormal rows)

SVD reveals the fundamental structure of any matrix and enables:

Low-rank approximation (keeping only largest singular values)
Image compression
Pseudoinverse computation
Dimensionality reduction

Example: Truncating an image's SVD to keep only the 50 largest singular values can reduce storage by 90% while maintaining recognizable quality.

What is Principal Component Analysis (PCA)?

PCA is a technique that finds the directions of maximum variance in data by computing eigenvectors of the covariance matrix. The first principal component points in the direction of greatest variance, the second in the direction of greatest remaining variance (perpendicular to the first), and so on.

PCA is used for:

Dimensionality reduction (keeping top k components)
Data visualization (projecting high-dimensional data to 2D or 3D)
Feature extraction (finding the most informative directions)
Noise reduction (removing low-variance components)

Example: Applying PCA to face images produces "eigenfaces"—the principal components that capture the most variation in facial appearance.

How do neural networks use linear algebra?

Neural networks are fundamentally composed of linear algebra operations:

Weight matrices connect layers through matrix-vector multiplication
Bias vectors add constant offsets to layer outputs
Forward propagation chains matrix multiplications with nonlinear activations
Backpropagation uses chain rule with Jacobian matrices to compute gradients
Batch processing uses matrix-matrix multiplication for efficiency

Each layer computes: output = activation(W·input + b) where W is the weight matrix and b is the bias vector.

Example: A layer connecting 100 inputs to 50 outputs uses a 50×100 weight matrix containing 5,000 learnable parameters.

What is the attention mechanism in transformers?

The attention mechanism computes weighted combinations of values based on the relevance between queries and keys. Given Query (Q), Key (K), and Value (V) matrices:

\[\text{Attention}(Q, K, V) = \text{softmax}\left(\frac{QK^T}{\sqrt{d_k}}\right)V\]

This allows each position in a sequence to "attend to" relevant positions elsewhere. The dot product Q·K^T measures similarity, softmax normalizes to weights, and the weighted sum of V produces the output.

Multi-head attention runs multiple attention operations in parallel, allowing the model to attend to different types of relationships simultaneously.

What is a Kalman filter?

A Kalman filter is an optimal algorithm for estimating the state of a system from noisy measurements. It works in two steps:

Predict: Use a system model to predict the next state
Update: Correct the prediction using new measurements

The Kalman gain determines how much to trust the prediction versus the measurement. The filter optimally combines both sources of information based on their uncertainties.

Example: A GPS receiver uses Kalman filtering to estimate position by fusing satellite measurements (accurate but slow) with inertial sensors (fast but drifts).

Technical Detail Questions

What is the difference between L1, L2, and L-infinity norms?

These are different ways to measure vector length:

Norm	Formula	Geometric Interpretation
L1 (Manhattan)	$\|v\|_1 = \sum	v_i
L2 (Euclidean)	$\\|v\\|_2 = \sqrt{\sum v_i^2}$	Straight-line distance
L∞ (Max)	$\|v\|_\infty = \max	v_i

Example: For vector (3, -4), L1 = 7, L2 = 5, L∞ = 4.

What is the difference between a symmetric and orthogonal matrix?

A symmetric matrix equals its own transpose: A = A^T. This means the element in row i, column j equals the element in row j, column i. Symmetric matrices have real eigenvalues and orthogonal eigenvectors.

An orthogonal matrix has columns (and rows) that are orthonormal: Q^TQ = QQ^T = I. This means Q^(-1) = Q^T. Orthogonal matrices preserve lengths and angles—rotations and reflections are orthogonal.

Example: Covariance matrices are symmetric. Rotation matrices are orthogonal.

What is the difference between rank and nullity?

The rank of a matrix is the dimension of its column space—the number of linearly independent columns. The nullity is the dimension of its null space—the number of independent vectors that map to zero.

The Rank-Nullity Theorem states: rank(A) + nullity(A) = number of columns.

Example: A 3×5 matrix with rank 3 has nullity 5 - 3 = 2, meaning two free variables exist in the solution to Ax = 0.

What is the condition number and why does it matter?

The condition number of a matrix is the ratio of its largest to smallest singular value. It measures how sensitive solutions are to small changes in input:

Condition number ≈ 1: Well-conditioned (stable)
Condition number > 10^10: Ill-conditioned (numerically unstable)

An ill-conditioned matrix amplifies rounding errors, potentially making computed solutions unreliable.

Example: A matrix with condition number 10^6 can amplify input errors by up to a million times in the output.

What is the difference between row echelon form and reduced row echelon form?

Row echelon form (REF): - Leading entries (pivots) are 1 - Each pivot is to the right of the pivot above - Rows of all zeros are at the bottom - Entries below each pivot are zero

Reduced row echelon form (RREF) adds: - Each pivot is the only nonzero entry in its column

RREF makes reading solutions easier but requires more computation to achieve.

What is the difference between QR and LU decomposition?

Feature	LU Decomposition	QR Decomposition
Form	A = LU (Lower × Upper triangular)	A = QR (Orthogonal × Upper triangular)
Matrix type	Square, some need pivoting	Any matrix
Stability	May need partial pivoting	Inherently stable
Primary use	Solving linear systems	Least squares, eigenvalue algorithms
Computation	Generally faster	More stable for ill-conditioned problems

What is the pseudoinverse?

The pseudoinverse A^+ generalizes matrix inversion to non-square and singular matrices. It's computed from SVD as:

\[A^+ = V\Sigma^+U^T\]

where Σ^+ is formed by taking reciprocals of nonzero singular values.

The pseudoinverse solves least squares problems: x = A^+b minimizes ||Ax - b||.

What is the difference between gradient descent and Newton's method?

Feature	Gradient Descent	Newton's Method
Uses	First derivatives (gradient)	First and second derivatives (Hessian)
Step direction	Steepest descent	Newton step using curvature
Convergence	Linear (slow near minimum)	Quadratic (fast near minimum)
Per-iteration cost	Low (gradient only)	High (Hessian inversion)
Robustness	Works far from minimum	May diverge far from minimum

For large-scale problems, quasi-Newton methods like BFGS approximate the Hessian without computing it explicitly.

What is the difference between convolution and correlation in image processing?

Convolution flips the kernel before sliding it across the image. Correlation does not flip the kernel. For symmetric kernels, they're identical.

Mathematically, convolution is associative (order doesn't matter for multiple filters), which is important for neural network design.

In practice, most deep learning frameworks implement correlation but call it "convolution."

What are homogeneous coordinates?

Homogeneous coordinates add an extra dimension to represent points. A 2D point (x, y) becomes (x, y, 1) in homogeneous coordinates. This enables:

Representing translations as matrix multiplication
Unified treatment of affine and projective transformations
Representing points at infinity
Simplifying perspective projection

To convert back: (x, y, w) → (x/w, y/w)

Example: Translation, which is not a linear transformation in Cartesian coordinates, becomes a matrix multiplication in homogeneous coordinates.

Common Challenge Questions

Why do I get different results when multiplying matrices in different orders?

Matrix multiplication is not commutative: AB ≠ BA in general. The order matters because:

A applies to the result of B, not the other way around
Dimensions may not even allow reverse multiplication
Geometrically, applying transformation A then B differs from B then A

Example: Rotating then scaling gives a different result than scaling then rotating.

How do I know if a system of equations has a unique solution, no solution, or infinitely many solutions?

Analyze the augmented matrix after row reduction:

Condition	Solution Type
Pivot in every column (of coefficient matrix)	Unique solution
Pivot in last column (constant column)	No solution
Fewer pivots than variables	Infinitely many solutions

The rank of the coefficient matrix compared to the augmented matrix determines solvability.

Why does my matrix inversion give numerical errors?

Numerical errors in matrix inversion occur when:

Matrix is singular or near-singular: Small pivots cause division by tiny numbers
Poor conditioning: Large condition number amplifies rounding errors
Accumulated errors: Long computation chains compound small errors

Solutions: - Use LU or QR decomposition instead of explicit inversion - Apply partial pivoting - Use higher precision arithmetic for critical applications - Reformulate the problem to avoid explicit inversion

How do I handle non-square matrices?

Non-square matrices can't be inverted directly, but you can:

For m×n with m > n (overdetermined): Use pseudoinverse or least squares
For m×n with m < n (underdetermined): Solution has free variables; use minimum norm solution
For any case: SVD works on all matrices and provides the pseudoinverse

Why do eigenvalue computations sometimes give complex numbers?

Complex eigenvalues occur when a real matrix includes rotational components. For example, a pure rotation matrix in 2D has eigenvalues cos(θ) ± i·sin(θ).

Complex eigenvalues always come in conjugate pairs for real matrices. They indicate oscillatory behavior in dynamical systems.

How do I choose the right matrix decomposition?

Problem	Best Decomposition
Solve Ax = b (general)	LU with pivoting
Solve Ax = b (symmetric positive definite)	Cholesky
Least squares	QR
Eigenvalues/vectors	Use specialized eigenvalue algorithms
Dimensionality reduction	SVD or eigendecomposition
Low-rank approximation	Truncated SVD
Numerical stability critical	QR or SVD

Why is gradient descent slow for some problems?

Gradient descent can be slow when:

Ill-conditioning: Different dimensions have very different scales
Saddle points: Gradient is small but not at a minimum
Plateaus: Loss surface is nearly flat
Learning rate issues: Too small = slow; too large = oscillation

Solutions: Use adaptive methods (Adam, RMSprop), apply preconditioning, or normalize features.

How do I debug dimension mismatch errors in neural networks?

Common dimension mismatch causes:

Matrix multiplication: Inner dimensions must match (m×n times n×p)
Batch dimension confusion: First dimension is usually batch size
Flattening errors: Wrong reshape before fully connected layers
Convolution output: Calculate output size using (input - kernel + 2×padding)/stride + 1

Trace dimensions through each layer systematically to find the mismatch.

Best Practice Questions

When should I use sparse matrix representations?

Use sparse matrices when:

More than 90% of entries are zero
Matrix is large (thousands of rows/columns)
Memory is constrained
Operations preserve sparsity

Common sparse formats include CSR (fast row slicing), CSC (fast column slicing), and COO (fast construction).

Example: A 10,000×10,000 matrix with only 50,000 nonzero entries uses 99.95% less memory in sparse format.

How do I choose the number of principal components to keep?

Common approaches:

Variance threshold: Keep components explaining 95% or 99% of total variance
Scree plot: Look for an "elbow" where variance explained drops sharply
Cross-validation: Choose k that minimizes reconstruction error on held-out data
Domain knowledge: Keep components that have interpretable meaning

There's no universal rule—the best choice depends on your specific application.

What regularization strength should I use?

Finding the right regularization strength (λ) typically requires:

Cross-validation: Try multiple values and evaluate on validation set
Grid search: Systematically explore a range (often logarithmic: 0.001, 0.01, 0.1, 1, 10)
Domain knowledge: Larger λ when you expect simpler relationships
Monitoring: Watch for underfitting (λ too large) or overfitting (λ too small)

Start with λ = 1 and adjust based on validation performance.

How should I normalize features before applying linear algebra algorithms?

Common normalization strategies:

Method	Formula	When to Use
Standardization	(x - μ) / σ	Features with different scales; PCA
Min-Max	(x - min) / (max - min)	Bounded output needed (0-1)
L2 Normalization	x /
Batch Normalization	Layer-wise during training	Deep neural networks

Always apply the same transformation to training and test data.

How do I handle missing data in matrix computations?

Options for missing data:

Imputation: Fill with mean, median, or predicted values
Matrix completion: Use low-rank methods to estimate missing entries
Mask and ignore: Weight valid entries only in loss computation
Drop rows/columns: If missingness is sparse and random

For recommendation systems, matrix completion methods specifically designed for sparse matrices work well.

What's the best way to implement matrix operations efficiently?

For efficient matrix operations:

Use optimized libraries: NumPy, BLAS, LAPACK, cuBLAS for GPU
Avoid explicit loops: Vectorize operations
Consider memory layout: Row-major vs column-major affects cache performance
Batch operations: Process multiple inputs simultaneously
Exploit structure: Use specialized algorithms for symmetric, sparse, or banded matrices
Avoid unnecessary copies: Use in-place operations when possible

How should I choose between different attention mechanisms?

Mechanism	Complexity	Best For
Dot-product	O(n²d)	Standard transformer, moderate sequences
Multi-head	O(n²d)	Learning multiple relationship types
Linear	O(nd²)	Very long sequences
Sparse	O(nd)	Extremely long sequences with local patterns
Cross-attention	O(nm d)	Different-length source and target

For most applications, multi-head dot-product attention works well.

Advanced Topic Questions

How does LoRA reduce the cost of fine-tuning large language models?

Low-Rank Adaptation (LoRA) decomposes weight updates as the product of two small matrices: ΔW = BA where B is (d × r) and A is (r × k) with rank r << min(d, k).

Instead of updating millions of parameters in the original weight matrix, LoRA only trains the small A and B matrices. This reduces:

Trainable parameters by 10-100x
Memory requirements during training
Storage for multiple fine-tuned models (only store A and B)

Example: For a 1000×1000 weight matrix, using rank r=8 reduces parameters from 1,000,000 to 16,000.

What is the relationship between SVD and eigendecomposition?

For a matrix A:

A^T A has eigenvalues σ² (squared singular values) and eigenvectors V
A A^T has eigenvalues σ² and eigenvectors U
The singular values of A are the square roots of eigenvalues of A^T A

For symmetric matrices, SVD and eigendecomposition are essentially the same, with singular values being absolute values of eigenvalues.

How does convolution in neural networks differ from mathematical convolution?

Aspect	Mathematical Convolution	Neural Network "Convolution"
Kernel flip	Yes	No (technically correlation)
Dimensions	Continuous or 1D discrete	2D, 3D with channels
Kernel	Fixed	Learned from data
Goal	Signal processing	Feature extraction

The terminology is historical—neural network convolution is mathematically cross-correlation, but the learned kernels make the distinction practically irrelevant.

How do quaternions avoid gimbal lock?

Gimbal lock occurs with Euler angles when two rotation axes align, losing a degree of freedom. Quaternions avoid this by:

Representing rotations as 4D unit vectors (avoiding singularities)
Using a different mathematical structure without axis-angle limitations
Enabling smooth interpolation (SLERP) between orientations

The trade-off is less intuitive representation—quaternion components don't directly correspond to roll, pitch, yaw angles.

What makes SLAM computationally challenging?

SLAM (Simultaneous Localization and Mapping) is challenging because:

Chicken-and-egg problem: Need position to build map, need map to determine position
Growing state: Map size grows over time, increasing computation
Loop closure: Recognizing previously visited locations requires global matching
Real-time constraints: Must process sensor data fast enough for navigation
Uncertainty management: Probabilistic state estimation with correlated errors

Modern SLAM systems use sparse representations, keyframe-based approaches, and graph optimization to manage complexity.

How do I design a custom loss function using matrix operations?

When designing custom loss functions:

Express in matrix form: Vectorize to enable batch computation
Ensure differentiability: Gradient must exist for backpropagation
Consider numerical stability: Avoid log(0), division by zero
Check convexity: Convex losses are easier to optimize
Match the problem: Classification → cross-entropy; regression → MSE or Huber

Example: Weighted least squares: L = (y - Xw)^T D (y - Xw) where D is a diagonal weight matrix.

What are the trade-offs between different sensor fusion approaches?

Approach	Pros	Cons
Kalman Filter	Optimal for linear Gaussian	Assumes linearity and Gaussian noise
Extended Kalman	Handles nonlinearity	Linearization errors, may diverge
Particle Filter	Any distribution	Computationally expensive
Factor Graph	Handles complex relationships	Complex implementation
Neural Fusion	Learns optimal fusion	Requires training data, less interpretable

Choose based on your system's characteristics and computational constraints.

How can I verify my linear algebra implementations are correct?

Testing strategies:

Identity checks: Does multiplying by identity return input?
Inverse checks: Does AA^(-1) = I?
Orthogonality checks: Is Q^T Q = I for orthogonal Q?
Numerical comparison: Compare with NumPy/SciPy results
Known solutions: Test on problems with analytical solutions
Property preservation: Do transformations preserve expected properties?
Gradient checking: Compare analytical gradients with numerical differences

What emerging applications of linear algebra should I be aware of?

Active areas applying linear algebra include:

Quantum computing: Quantum states are vectors; operations are unitary matrices
Graph neural networks: Message passing as sparse matrix operations
Neural radiance fields (NeRF): 3D scene representation using transformations
Diffusion models: Noise addition/removal as matrix operations
Mixture of experts: Sparse gating with linear combinations
State space models: Efficient alternatives to attention using matrix structure

Understanding linear algebra fundamentals prepares you for these advancing fields.