What does this Linear Algebra cheat sheet cover?

It has 11 sections: Row Reduction, Matrices, Determinants, Vector Spaces, Subspaces, Rank-Nullity, Eigenvalues, Diagonalization, Orthogonality, Least Squares and Invertible Theorem. Together they cover the topics a typical Linear Algebra exam tests, including row reduction and RREF, matrix operations and inverses, determinants and properties and vector spaces, span, basis, dimension.

Is the linear algebra cheat sheet free?

The first part is free to read on this page, with no account needed. The full sheet opens in the app with Unlimited ($4.99 a month or $19 for a semester), where you can edit it, print it or save it as a PDF. It is one US Letter page and prints cleanly in black and white or colour.

Can I bring this Linear Algebra cheat sheet to my exam?

Only if your course allows a note sheet, so check the syllabus. This one is a single US Letter page. If your exam allows a different size, such as a 3x5 or 4x6 index card or one page front and back, Cheat Sheet AI makes one at exactly that size from your own lecture notes.

The sheet was written by our engine from a standard course outline, then a second model fact-checked every formula and statement against it; it corrected 1 statement before publishing. Notation and emphasis vary between courses, so check it against your own lecture notes.

Can I make a Linear Algebra sheet from my own notes?

Yes. Upload your lecture slides, notes or study guide and Cheat Sheet AI writes a sheet in your course's notation, sized to exactly the space your exam allows. Your first sheets are free.

MathLinear Algebra

Linear Algebra Cheat Sheet — 1-page PDF

Row reduction, vector spaces, determinants, eigenvalues, orthogonality, LSQ. The whole course on one dense, printable page for your midterm or final, with every formula typeset and worked examples where they help.

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For a specific exam: Linear Algebra Midterm study sheet, Linear Algebra Final exam study sheet.

1 pageUS Letter
11 sections11 topics
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Preview of the one-page Linear Algebra cheat sheet: Row reduction, vector spaces, determinants, eigenvalues, orthogonality, LSQ — the whole course, one page

On the sheet

What’s on it

The topics a typical Linear Algebra exam tests, all on the one page.

Row reduction and RREF
Matrix operations and inverses
Determinants and properties
Vector spaces, span, basis, dimension
Rank-nullity theorem
Column space and null space
Eigenvalues and eigenvectors
Diagonalization
Orthogonality, Gram-Schmidt, projections
Least squares
Invertible matrix theorem

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The free part, as text

The free part of the sheet, section by section. Unlock the full sheet to read the rest, edit any block, and print it or save it as a PDF.

01Row Reduction

Row echelon form: Leading entries step right as you go down; zero rows at bottom; entries below each leading entry are $0$ .
Reduced row echelon form: REF + each leading entry is $1$ and is the only nonzero entry in its column.

Use elementary row ops: swap rows; multiply row by nonzero scalar; add multiple of one row to another.
Pivot column = column containing a leading $1$ in RREF.
Free variable = nonpivot variable.

Row equivalence

$A \sim B ⟺ B$ can be obtained from $A$ by elementary row operations

Row ops preserve solution set of $A x = b$ .

Ex.Find pivots from RREF

If $rref (A) = 100010 2 - 3 0$ , pivots are in cols $1, 2$ ; rank $= 2$ ; col $3$ free.

Watch out:Do not read pivot positions from the original matrix; use the RREF / echelon form.

02Determinants

Triangular

det A = \prod_{i} a_{ii}

If $A$ is upper/lower triangular.

Cofactor expansion

det A = \sum_{j = 1}^{n} a_{ij} C_{ij} = \sum_{i = 1}^{n} a_{ij} C_{ij}

Expand along any row $i$ or column $j$ . $C_{ij} = (- 1)^{i + j} M_{ij}$ .

Determinant tests

Condition	Meaning
$det A \neq = 0$	$A$ invertible
$det A = 0$	$A$ singular
one zero row	$det A = 0$

Watch out:When using row reduction to compute $det A$ , track the sign/scale changes from row ops.

03Vector Spaces

Vector space: Set $V$ with addition and scalar multiplication satisfying the vector space axioms.
Span: $span {v_{1}, \dots, v_{k}} = {c_{1} v_{1} + \dots + c_{k} v_{k}}$ .
Linear independence: ${v_{i}}$ independent iff $c_{1} v_{1} + \dots + c_{k} v_{k} = 0$ implies all $c_{i} = 0$ .
Basis: A linearly independent spanning set.

04Rank-Nullity

Pivot count

$rank (A) = #$ pivot columns $= #$ pivot rows

In RREF / echelon form.

Invertible size

rank (A) = n ⟺ N (A) = {0}

For $A$ $m \times n$ ; if square, this is one invertibility test.

Rank-nullityIf $A$ is $5 \times 8$ and $rank (A) = 3$ , find nullity.

$nullity (A) = 8 - 3 = 5$ .

Dimensions

Space	Dimension
$N (A)$	$n - rank (A)$
$Col (A)$	$rank (A)$
$Row (A)$	$rank (A)$

Watch out:Nullity uses the number of columns $n$ , not rows.

05Eigenvalues

Eigenvalue / eigenvector: $A v = λ v$ with $v \neq = 0$ . Then $λ$ is an eigenvalue and $v$ an eigenvector.

Characteristic equation

det (A - λ I) = 0

Eigenvalues are roots of the characteristic polynomial.

06Orthogonality

Inner product: In $R^{n}$ , $u \cdot v = u^{T} v$ . Orthogonal iff $u \cdot v = 0$ .

Norm / distance

∥ v ∥ = v \cdot v, d (u, v) = ∥ u - v ∥

Projection

proj_{u} v = \frac{v \cdot u}{u \cdot u} u

Projection of $v$ onto nonzero $u$ .

Orthogonal set

{u_{i}} orthogonal ⟺ u_{i} \cdot u_{j} = 0 (i \neq = j)

If nonzero, orthogonal set is linearly independent.

Gram-Schmidt

u_{1} = v_{1}, u_{k} = v_{k} - \sum_{j = 1}^{k - 1} proj_{u_{j}} v_{k}

Produces orthogonal $u_{k}$ ; normalize to get orthonormal basis.

Watch out:Projection formula needs $u \neq = 0$ ; if $u$ is not unit, keep denominator $u \cdot u$ .

07Least Squares

Least-squares solution: Minimizes $∥ A x - b ∥$ when $A x = b$ has no exact solution.

Normal equations

A^{T} A x = A^{T} b

Least-squares solutions satisfy this.

Residual

r = b - A \overset{x}{^}

At solution, $r ⊥ Col (A)$ .

Cheat sheet locked

Also on the full Linear Algebra sheet

9 more sections, with every definition, formula and worked example. Unlimited unlocks every full sheet in the library to edit, print or save as a PDF, and makes unlimited sheets from your own notes.

Matrices

Product
Inverse
Transpose rules
Inverse rules
Fast facts

Determinants

2x2
Properties
Row ops

Vector Spaces

Dimension

Subspaces

Subspace test
Column space
Null space
Rank
Nullity
How to find bases

Rank-Nullity

Rank-nullity

Eigenvalues

Eigenspace
2x2 trace/det

Diagonalization

Diagonalizable
Build
Power rule
Criterion

Least Squares

Projection view
Methods

Invertible Theorem

Equivalent conditions
Invertible matrix theorem

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FAQ

Questions

About the Linear Algebra sheet and making your own.

What does this Linear Algebra cheat sheet cover?: It has 11 sections: Row Reduction, Matrices, Determinants, Vector Spaces, Subspaces, Rank-Nullity, Eigenvalues, Diagonalization, Orthogonality, Least Squares and Invertible Theorem. Together they cover the topics a typical Linear Algebra exam tests, including row reduction and RREF, matrix operations and inverses, determinants and properties and vector spaces, span, basis, dimension.
Is the linear algebra cheat sheet free?: The first part is free to read on this page, with no account needed. The full sheet opens in the app with Unlimited ($4.99 a month or $19 for a semester), where you can edit it, print it or save it as a PDF. It is one US Letter page and prints cleanly in black and white or colour.
Can I bring this Linear Algebra cheat sheet to my exam?: Only if your course allows a note sheet, so check the syllabus. This one is a single US Letter page. If your exam allows a different size, such as a 3x5 or 4x6 index card or one page front and back, Cheat Sheet AI makes one at exactly that size from your own lecture notes.
How accurate is it?: The sheet was written by our engine from a standard course outline, then a second model fact-checked every formula and statement against it; it corrected 1 statement before publishing. Notation and emphasis vary between courses, so check it against your own lecture notes.
Can I make a Linear Algebra sheet from my own notes?: Yes. Upload your lecture slides, notes or study guide and Cheat Sheet AI writes a sheet in your course's notation, sized to exactly the space your exam allows. Your first sheets are free.

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