Linear algebra first course study map

The goal of this pathway is to see matrices as geometric machines, not only as grids of numbers to grind through. It orders the linear algebra pages, glossary entries, and 2 by 2 calculators that already live on Higher Math Guide. It is a first course map, educational only, and not a substitute for a syllabus.

Who this pathway 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 should be able to plot points and solve a tiny system by elimination. Calculus can wait until the Jacobian steps.

If the course is mostly proofs, pair this geometry with intro to proofs. If multivariable calculus is next, this map is the bridge before partial derivatives.

Prerequisites

You need algebra, coordinates in the plane, and the habit of checking a claim on a 2 by 2 example. If vector and matrix are new words, read vector, matrix, and span before the eigenvalue explainer. The sources page lists MIT OpenCourseWare and Gilbert Strang’s linear algebra materials for a full set of notes.

What you will be able to do

When you finish the core, you should read a dot product’s sign, name matrix columns as images of the standard basis, and say what a determinant does to area. You should decide whether a singular system has no solution or many. You should find 2 by 2 eigenvalues, check $A\mathbf{v}=\lambda\mathbf{v}$ by hand, and notice when a repeated eigenvalue still fails to give two independent eigenvectors. You should describe a Jacobian as the best linear approximation, not as a bag of partial derivatives.

Suggested 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 site wide rule.

1. Read the linear algebra topic as a table of contents for maps, bases, and eigenvalues.
2. Read vector, span, linear independence, basis, linear map, matrix, and determinant. Pair each word with a 2 by 2 picture.
3. Open the vector operations calculator for a dot product, a 3D cross product, a magnitude, or a unit vector. It does not print an angle. Zero means orthogonal (or a zero vector), positive means a similar direction, and negative means an obtuse angle. A cross product’s length is the parallelogram area.
4. Open the 2 by 2 matrix calculator for a determinant, an inverse when it exists, and the products $AB$ and $BA$. On paper first, read the columns as the images of the standard basis vectors. This page does not compute eigenvalues. A zero determinant means there is no inverse.
5. When the matrix is 3 by 3, open the 3 by 3 matrix calculator for cofactor expansion and an inverse in exact fractions. To see each elimination step of a linear system, open the reduced row echelon form calculator. Neither page finds eigenvalues.
6. Read eigenvalues in plain language, then eigenvalue, eigenvector, and eigenspace. An eigenvector is never the zero vector. The eigenvalue is the stretch along a line the map does not tip off.
7. Check roots on the 2 by 2 eigenvalue calculator. It reports trace, determinant, and the roots of the characteristic polynomial. It does not return eigenvectors, and it does not decide whether a repeated root gives two independent directions. You do that by hand in the checkpoint below. If the quadratic formula itself is rusty, the quadratic formula calculator is the same algebra without a matrix.
8. When a homework matrix is 3 by 3, open the 3x3 eigenvalue calculator. It expands $\det(A-\lambda I)$ from the trace, the principal minors, and the determinant, then row reduces for a real eigenvector. The pictures in this pathway stay 2 by 2.
9. Optional: read the multivariable and vector calculus intro and the gradient entry. A gradient is the vector of partial derivatives.
10. Read Jacobian matrices in plain language and the Jacobian entry. Fill the partial derivatives yourself, then use the 2 by 2 matrix calculator for a product or a determinant. That calculator multiplies two 2 by 2 matrices. It does not apply a matrix to a column.

Checkpoints and practice habits

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 the determinant on the matrix calculator.

Dot product check. Take $(1,2)$ and $(2,-1)$. The products $1\cdot 2$ and $2\cdot(-1)$ cancel, so the dot product is $0$ and the vectors are orthogonal. Confirm that zero on the vector operations calculator. The page will not print the angle.

Repeated eigenvalue check. Let $A=\begin{pmatrix}2&1\\0&2\end{pmatrix}$. It is triangular, so both eigenvalues are $2$. Trace $4$ and determinant $4$ give $\lambda^{2}-4\lambda+4=(\lambda-2)^{2}$. The eigenvalue calculator should agree, though a discriminant that ought to be zero can print as a tiny nonzero number. Treat that split as one repeated eigenvalue. Eigenvectors are your job: $A-2I=\begin{pmatrix}0&1\\0&0\end{pmatrix}$, so every eigenvector is a multiple of $(1,0)$. There is no second independent direction, and the matrix is not diagonalizable. The calculator cannot tell you that.

Singular system check. Let $A=\begin{pmatrix}1&2\\2&4\end{pmatrix}$. The determinant is $0$, so the matrix calculator should refuse an inverse. The second column is twice the first, so reachable outputs are multiples of $(1,2)$. Thus $\mathbf{b}=(3,6)$ works and $\mathbf{b}=(3,0)$ does not. A solvable singular system gives a line of solutions, not one point.

Common pitfalls for self learners

Complex eigenvalues are a rotation and a scale, with no real invariant line. Row reduction tells you which equations are redundant. The columns tell you what the map does. If the Jacobian sentence feels empty, read what a derivative actually measures on the AP Calculus AB bridge, then return.

The zero vector is not an eigenvector. It satisfies $A\mathbf{0}=\lambda\mathbf{0}$ for every $\lambda$, so the definition excludes it. The eigenspace puts zero back in, because a subspace must.

Related reading and the next pathway

If the next demand is writing, continue with intro to proofs. A fair first proof: two vectors are linearly dependent when one is a scalar multiple of the other. Define the words, state the assumption, and end by naming what you proved.

If several variables are next, stay with the multivariable intro and Jacobian matrices in plain language. Lagrange multipliers need gradients and a constraint, so they are later reading. If derivative language slowed the Jacobian step, the AP Calculus AB bridge repairs it. The three routes are compared on the pathways hub.