Linear algebra studies linear maps and the spaces they act on. A linear map sends lines through the origin to lines through the origin, and it respects addition and scaling. Matrices are concrete recipes for those maps once bases are chosen. Eigenvalues and eigenvectors reveal stretch along lines that coordinates can hide. This pillar is a first course map of those ideas, with a study order into the glossary, the explainers, and the calculators that already live here, including a 3 by 3 determinant and inverse.

Who this pillar is for

Start here if you are beginning linear algebra, or returning to vectors and matrices and wanting a picture before a pile of algorithms. You might be in a first matrix course, preparing for multivariable calculus, or trying to make eigenvalues mean something before a calculator prints them.

If you want the sittings spelled out, with extra checkpoints, use the linear algebra first course pathway after you read this map. If the course is mostly proofs, pair the geometry here with intro to proofs. Calculus can wait until the Jacobian steps.

Prerequisites

You need algebra, coordinates in the plane, and the habit of checking a claim on a 2 by 2 example. You should be able to plot a point and solve a tiny system by elimination. If vector, matrix, and span are new words, read those glossary lines before the eigenvalue explainer.

The sources page lists MIT OpenCourseWare and Gilbert Strang's linear algebra materials for a full set of notes. This page will not replace that course. It tells you which sentence to hold while you compute.

What you should be able to do

When you finish the core, you should be able to do the following on a 2 by 2 example, and say which words survive in higher dimensions.

  • Add vectors and read a dot product's sign: similar direction, right angle, or obtuse angle.
  • Describe a basis as a minimal building set, and coordinates as the recipe relative to that basis.
  • Read the columns of a matrix as the images of the standard basis, and say what a determinant does to area.
  • Decide, for a singular system, whether $A\mathbf{x}=\mathbf{b}$ has no solution or many.
  • Find 2 by 2 eigenvalues, check $A\mathbf{v}=\lambda\mathbf{v}$ by hand, and notice when a repeated eigenvalue fails to give two independent eigenvectors.
  • Describe a Jacobian as the best linear approximation to a map of several variables, when you are ready for that handoff.

Vectors, span, and bases

A vector is an arrow you can add and scale, or more abstractly an element of a vector space. The span of a collection is every linear combination of those vectors: all the places you can reach by adding and scaling them. Linear independence means the only combination that lands on zero uses all zero coefficients. A dependent set has a redundancy. One vector is a combination of the others, so it was not earning its place.

A basis is a linearly independent spanning set. Every vector has unique coordinates relative to a basis. Change the basis and the numbers change, even though the geometric vector is the same. That is why a matrix is always "relative to chosen bases," as the matrix glossary line says. The arrow did not change. The recipe did.

Picture. In the plane, $(1,0)$ and $(0,1)$ form a basis. So do $(1,1)$ and $(1,-1)$, because they are not parallel and they span the plane. The vector $(2,0)$ has coordinates $(2,0)$ in the first basis and $(1,1)$ in the second, since $(2,0)=(1,1)+(1,-1)$. Same arrow, different recipe.

Matrices as machines

Multiplying by a matrix $A$ applies a linear map. The columns of $A$ are the images of the standard basis vectors. If you know where the basis goes, you know the whole map, because every other vector is a combination of the basis and the map respects combinations.

Solving $A\mathbf{x}=\mathbf{b}$ asks which inputs produce $\mathbf{b}$. Row reduction tells you which equations are redundant. The columns tell you what the map does. Use both stories. The reduced row echelon form calculator writes those elementary row operations and tells you whether the system has one solution, infinitely many, or none.

The determinant of a 2 by 2 real matrix is the signed area scale of the parallelogram formed by the columns. Positive preserves orientation. Negative flips it. Zero means the columns are parallel, the map flattens the plane, and there is no inverse. Then $A\mathbf{x}=\mathbf{b}$ has either no solution or a whole line of them, depending on whether $\mathbf{b}$ is in the column span.

Let $A=\begin{pmatrix}2&4\\1&2\end{pmatrix}$. The second column is twice the first, so the determinant is $0$. Reachable outputs are multiples of $(2,1)$. Thus $\mathbf{b}=(2,1)$ has solutions and $\mathbf{b}=(1,0)$ does not. The 2 by 2 matrix calculator should refuse an inverse. Predict that refusal before you type the entries. The page also multiplies two matrices. It does not compute eigenvalues. When the matrix is 3 by 3, the 3 by 3 matrix calculator expands the determinant along the first row and builds the inverse from the adjugate, in exact fractions.

Eigenvalues: lines the map keeps

An eigenvector is a nonzero vector $\mathbf{v}$ with $A\mathbf{v}=\lambda\mathbf{v}$. The map stretches or flips $\mathbf{v}$ and does not tip it off its line. The scalar $\lambda$ is the eigenvalue. The zero vector satisfies the equation for every $\lambda$, so the definition excludes it. The eigenspace puts zero back in, because a subspace must contain zero. Read eigenvalues in plain language for the geometric tour, then the glossary lines for eigenvalue and eigenvector.

For a 2 by 2 matrix the eigenvalues are the roots of a quadratic. The trace and the determinant build that quadratic: $\lambda^{2}-(\operatorname{tr} A)\lambda+\det A$. The 2 by 2 eigenvalue calculator reports the trace, the determinant, and the roots. It does not return eigenvectors, and it does not decide whether a repeated root gives two independent directions. You do that by hand. If the quadratic formula itself is rusty, the quadratic formula calculator is the same algebra without a matrix.

For a 3 by 3 matrix the characteristic polynomial is a cubic. The 3x3 eigenvalue calculator expands $\det(A-\lambda I)$ from the trace, the sum of the principal 2 by 2 minors, and the determinant, then row reduces $A-\lambda I$ for a real eigenvector.

Check. Let $A=\begin{pmatrix}1&1\\0&2\end{pmatrix}$. It is triangular, so the eigenvalues are the diagonal entries $1$ and $2$. Trace $3$ and determinant $2$ give $\lambda^{2}-3\lambda+2=(\lambda-1)(\lambda-2)$. For $\lambda=1$, $A-I=\begin{pmatrix}0&1\\0&1\end{pmatrix}$, so eigenvectors are multiples of $(1,0)$. For $\lambda=2$, $A-2I=\begin{pmatrix}-1&1\\0&0\end{pmatrix}$, so eigenvectors are multiples of $(1,1)$. Check $A(1,1)=(2,2)$. The eigenvalue calculator should agree about the roots. The eigenvectors are your job.

Where the geometry goes next

A gradient is a vector of partial derivatives. A Jacobian is the matrix of them, the best linear approximation to a map at a point. Read multivariable calculus and Jacobian matrices in plain language after eigenvalues feel like lines. Fill the partials yourself. Use the matrix calculator only for a determinant you have already set up.

Not every matrix is diagonalizable. A repeated eigenvalue may give only one independent direction. Complex eigenvalues of a real matrix mix a rotation and a scale: there is no real invariant line. Learn the stretch story first. The special cases are easier once that picture is drawable.

Suggested learning order

Work in this order. Open a calculator only after you can say what the number means. The FAQ on calculators versus understanding is the rule on this site. The linear algebra first course follows the same arc with more checks. Use that pathway if you want the longer route.

  1. Glossary, each with a 2 by 2 picture: vector, span, linear independence, basis, matrix, determinant.
  2. The vector operations calculator for a dot product you already computed. Zero means a right angle or a zero vector. The page does not print an angle.
  3. The Gram Schmidt calculator when a list of vectors should become an orthogonal basis. Order matters, and a zero remainder means dependence.
  4. The LU decomposition calculator when elimination should keep the multipliers instead of clearing all the way to reduced form.
  5. The 2 by 2 matrix calculator for a determinant and for $AB$ versus $BA$, after you read the columns as images of the basis.
  6. The 3 by 3 matrix calculator for cofactor expansion, the adjugate, and an inverse in exact fractions.
  7. Eigenvalues in plain language, then the triangular example above by hand.
  8. Roots on the 2 by 2 eigenvalue calculator. Eigenvectors by hand for that case. When the matrix is 3 by 3, the 3x3 eigenvalue calculator row reduces for a real eigenvector. A repeated root still needs enough independent directions. The pathway has that checkpoint.
  9. When several variables are next: multivariable calculus and Jacobian matrices.

Checkpoints and practice habits

Practice habit. For every 2 by 2 matrix, draw the columns as arrows from the origin and say where the standard basis went. Then say what happened to area, using the sign and size of the determinant. Do this before you row reduce anything.

Area check. Let $A=\begin{pmatrix}2&0\\0&3\end{pmatrix}$. The columns $(2,0)$ and $(0,3)$ are the images of $(1,0)$ and $(0,1)$. The determinant is $6$, so oriented areas scale by $6$ and orientation is preserved. Confirm on the matrix calculator.

Independence check. Are $(1,2)$ and $(2,4)$ independent? The second is twice the first, so a combination with coefficients $2$ and $-1$ gives zero. They do not form a basis of the plane. Their span is one line.

Eigenvector check. For the triangular matrix above, $A(1,0)=(1,0)$ and $A(1,1)=(2,2)$. If a proposed eigenvector fails this multiplication test, it is not an eigenvector, whatever a solver printed.

Common pitfalls

Pitfall. Do not rush to the determinant as "the formula." For a 2 by 2 real matrix it is the signed area scale. Zero means the map flattens the plane, so there is no inverse, and $A\mathbf{x}=\mathbf{b}$ cannot have exactly one solution. Learn that sentence before the formula. A second trap: the zero vector is never an eigenvector, even though $A\mathbf{0}=\lambda\mathbf{0}$ is true for every $\lambda$.

Matrix multiplication is not commutative. Compute one product by hand, then compare $AB$ and $BA$ on the matrix calculator. If a Jacobian sentence feels empty, read what a derivative actually measures, then return.

Related reading

Follow the linear algebra first course for the sittings. Partial derivatives lead to multivariable calculus. Writing arguments lead to intro to proofs. A fair first proof: two vectors are linearly dependent when one is a scalar multiple of the other. Routes are compared on the pathways hub.

Read the eigenvalue explainer once columns are pictures. Open the eigenvalue calculator only after you can say what a root means.

Read eigenvalues in plain language
Disclaimer. Educational only. This page is not a course and not a substitute for a text. See the disclaimer.