Welcome to the linear algebra calculators hub — a growing collection of free, interactive tools for the matrix and vector mathematics that sits at the heart of machine learning. Every calculator shows step-by-step working, runs in your browser with no sign-up, and is paired with a plain-English guide so you learn the idea, not just the answer.
Why linear algebra matters for machine learning
If calculus is the language of how models learn, linear algebra is the language of how they represent data. A dataset is a matrix; an image is a matrix; the weights of a neural network are matrices; a word embedding is a vector. Training a model means transforming these objects with operations like multiplication, transposition, and decomposition — the very operations these calculators perform. Get comfortable with them and the rest of machine learning becomes far less mysterious.
This hub is organised the way the subject actually builds up: first the core matrix operations, then vector operations, then the decompositions and eigen-analysis that power dimensionality reduction and beyond. Work through them in order, or jump straight to the tool you need.
Matrix operation calculators
Matrices are rectangular grids of numbers, and almost everything in applied linear algebra starts with combining them. The foundational operation is matrix multiplication, where each entry of the result is the dot product of a row and a column — the exact computation performed inside every dense layer of a neural network. From there you build to the determinant, a single number that tells you whether a matrix can be inverted and how it scales space, and the transpose, which flips a matrix across its diagonal and appears constantly in gradient derivations.
Use the matrix multiplication calculator to multiply two matrices of any compatible size, the matrix determinant calculator to find the determinant of any 2×2 to 4×4 matrix by cofactor expansion, and the matrix transpose calculator to flip a matrix across its diagonal — each with worked steps. More matrix tools, including the inverse and rank, are on the way.
Vector operation calculators
Vectors are the workhorses of machine learning: a single data point, a gradient, an embedding, or a direction in space is a vector. The two operations you will reach for most are the dot product, which measures how aligned two vectors are and underpins cosine similarity, and the cross product, which produces a vector perpendicular to two others. The dot product in particular is the bridge between geometry and machine learning — it is how models measure similarity between an input and a learned pattern.
Try the dot product calculator, which also reports the angle between the two vectors, and the cross product calculator for 3D vectors — each with the geometry explained alongside the arithmetic.
Decompositions and eigen-analysis
The deepest ideas in applied linear algebra are the decompositions — ways of breaking a matrix into simpler factors that reveal its structure. Eigenvalues and eigenvectors describe the directions a transformation stretches and by how much; they are the mathematical core of Principal Component Analysis, the most widely used dimensionality-reduction technique. Singular Value Decomposition generalises the idea to any matrix and drives everything from recommender systems to data compression. These tools are on the roadmap for this hub and will slot in beside the matrix and vector calculators above.
How to get the most from these calculators
Each tool is built to teach, not just to compute. When you use one, do three things. First, predict the answer’s shape before you press the button — knowing that a 2×3 matrix times a 3×2 matrix yields a 2×2 result is often more valuable than the numbers themselves. Second, open the step panel and follow one entry all the way through, so the rule sticks. Third, reproduce the result in Python with NumPy; every guide includes the one-line NumPy equivalent, because in practice you will compute with code, and knowing the exact function to call is half the battle.
Linear algebra rewards this kind of active practice. The operations are simple individually — a few additions and multiplications — but they compose into the powerful transformations that make modern machine learning work. These linear algebra calculators are here to make that practice fast, visual, and free.
Frequently asked questions
Are these linear algebra calculators free?
Yes. Every tool on this site is completely free, needs no account, and runs entirely in your browser.
Do the calculators show steps?
They do. Each one reveals the key working — the row-column products behind a matrix multiplication, for example — so you can learn the method, not just copy the result.
Which calculator should I start with?
Start with the matrix multiplication calculator. It is the single most important operation in machine learning and everything else builds on the same row-by-column dot-product idea.