Vector magnitude
$$\|\mathbf v\|=\sqrt{v_1^2+\cdots+v_n^2}$$
Measure Euclidean vector length.
High school · UniversityLinear algebra formulas, grouped by topic.
17 linear algebra formulas across 7 topics, free and without an account. Linear algebra formulas operate on vectors and matrices instead of single numbers. Dot and cross products measure alignment and area, determinants and inverses decide whether a system has a unique solution, and eigenvalues expose the directions a transformation merely stretches. Dimensions are the recurring condition: a product is only defined when the inner dimensions agree.
Every linear algebra formula on one page
Each entry links to its own page, where the symbols are defined, the conditions are stated, and the formula is substituted through a worked example. Levels covered here: High school, University.
Vectors
5 formulas in vectors.
Unit vector
$$\widehat{\mathbf v}=\frac{\mathbf v}{\|\mathbf v\|}$$
Keep a vector’s direction while changing its length to one.
High school · UniversityDot product and angle
$$\mathbf u\cdot\mathbf v=\sum u_iv_i=\|\mathbf u\|\|\mathbf v\|\cos\theta$$
Measure alignment and recover the angle between vectors.
High school · UniversityVector projection
$$\operatorname{proj}_{\mathbf u}\mathbf v=\frac{\mathbf v\cdot\mathbf u}{\mathbf u\cdot\mathbf u}\mathbf u$$
Find the component of $\mathbf v$ pointing along $\mathbf u$.
UniversityCross product
$$\mathbf u\times\mathbf v=\langle u_2v_3-u_3v_2,\ u_3v_1-u_1v_3,\ u_1v_2-u_2v_1\rangle$$
Produce a vector perpendicular to two three-dimensional vectors.
UniversityMatrices
5 formulas in matrices.
Matrix multiplication
$$(AB)_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}$$
Compose two linear transformations by row-column products.
High school · UniversityTwo-by-two determinant
$$\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc$$
Measure signed area scaling and test invertibility.
High school · UniversityTwo-by-two inverse
$$A^{-1}=\frac1{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$$
Undo an invertible two-by-two linear transformation.
High school · UniversityMatrix transpose
$$(A^T)_{ij}=A_{ji}$$
Turn matrix rows into columns.
High school · UniversityTrace
$$\operatorname{tr}(A)=\sum_{i=1}^{n}a_{ii}$$
Add the main-diagonal entries of a square matrix.
UniversitySystems
2 formulas in systems.
Inverse solution of a system
$$A\mathbf x=\mathbf b\implies\mathbf x=A^{-1}\mathbf b$$
Solve a square linear system using an inverse.
UniversityCramer’s rule
$$A\mathbf x=\mathbf b,\qquad x_i=\frac{\det(A_i)}{\det(A)}$$
Express each coordinate of a unique square-system solution using determinants.
High school · UniversityEigenvalues
2 formulas in eigenvalues.
Eigenvalue equation
$$A\mathbf v=\lambda\mathbf v$$
Identify directions a transformation only scales.
UniversityCharacteristic equation
$$\det(A-\lambda I)=0$$
Find eigenvalues as roots of a polynomial.
UniversityVector spaces
1 formula in vector spaces.
Rank-nullity theorem
$$\operatorname{rank}(A)+\operatorname{nullity}(A)=n$$
Split the input dimension into visible output directions and null-space directions.
UniversityLeast squares
1 formula in least squares.
Normal equations
$$A^TA\widehat{\mathbf x}=A^T\mathbf b$$
Find a vector whose model prediction is closest in squared distance to data.
UniversityOrthogonality
1 formula in orthogonality.
Gram–Schmidt step
$$\mathbf u_k=\mathbf v_k-\sum_{j<k}\operatorname{proj}_{\mathbf u_j}\mathbf v_k$$
Remove earlier vector directions to build an orthogonal basis.
UniversityKeep going with linear algebra
A formula sheet is a reference, not a method. Use the subject guide to see where these relationships come from, a calculator to check a substitution you are unsure about, and practice questions to find out whether you can actually retrieve them.
- Study linear algebra — the subject hub with lessons, worked methods, and common mistakes.
- Linear algebra calculator — enter your own values and compare the result with your working.
- Linear Algebra Practice — retrieval practice across every topic listed above.
- Math formula library — search all 215 formulas by name, symbol, or condition.
Formulas in the other subjects
Mathematics does not stop at a subject boundary, and neither do its formulas. Each subject page below lists its own relationships, conditions, and worked examples.
Arithmetic
Review the core arithmetic relationships, their restrictions, and worked substitutions.
Algebra
Review the core algebra relationships, their restrictions, and worked substitutions.
Geometry
Review the core geometry relationships, their restrictions, and worked substitutions.
Trigonometry
Review the core trigonometry relationships, their restrictions, and worked substitutions.
Precalculus
Review the core precalculus relationships, their restrictions, and worked substitutions.
Calculus
Review the core calculus relationships, their restrictions, and worked substitutions.
Statistics
Review the core statistics relationships, their restrictions, and worked substitutions.
Probability
Review the core probability relationships, their restrictions, and worked substitutions.
Differential Equations
Review the core differential equations relationships, their restrictions, and worked substitutions.
Discrete Math
Review the core discrete math relationships, their restrictions, and worked substitutions.