Problem-solving recipes

A step-by-step procedure for every problem type this course teaches — what to do first, what to do next, and how to tell which case you're in.

What are you being asked to do?

Start with what you're actually being asked to produce.

Given two vectors or lines, need a geometric fact about them (angle, orthogonality, intersection, cross/triple product) — one of the vectors- and-geometry recipes below.

Given a system of equations, need the solution — Gaussian elimination. Need to know whether a solution exists or is unique without fully solving — the existence-and-uniqueness recipe instead.

Given a matrix, need its inverse — row reduction with the identity bolted on. Need $A=LU$ or $PA=LU$ instead — the factorization recipes. Already have $A^{-1}$ and just need to solve $A\mathbf x=\mathbf b$ — the solve-with-known-inverse recipe, a single matrix-vector multiply.

Given a matrix, need a fact about a subspace it defines (null space, column space, a general subspace's dimension or a vector's coordinates) — one of the vector-spaces-and-subspaces recipes below.

Given a linear transformation, need its matrix — the standard-matrix recipe, or the relative-to-a-basis version if the problem specifies a non-standard basis.

Given vectors or a matrix, need orthogonality or a least-squares fit — the orthogonal-complement, QR, or normal-equations recipes below, depending on what's given and what's being solved for.

Given a matrix, need its eigenvalues — the characteristic equation (or the trace/determinant shortcut for a $2\times2$ ). Already have an eigenvalue and need its eigenvectors instead — the eigenspace recipe, not the characteristic equation again.

Given a matrix, asked for $P$ and $D$ with $A=PDP^{-1}$ — diagonalization, which itself calls the eigenvalue and eigenvector recipes above as its first two steps. Asked for $A^k$ specifically — the diagonalization-power recipe, which reuses a diagonalization you may have already done.

Readiness — trig, notation, functions, and proof

Find a triangle's angle from its three side lengths

Use when: you know all three side lengths of a triangle and need the angle opposite one of them, with no angle given directly

Goal: the measure of the angle opposite the side you named, without ever drawing the triangle to scale

Angles and cosine — the trig this course actually uses

Write an explicit sum in sigma notation

Use when: you have a sum written out term by term and need to compress it into one $\sum$ expression

Goal: a single $\sum$ expression — index letter, starting value, ending value, and general term — that expands back out to the exact sum you started with

Sigma notation and index bookkeeping

Find the domain of a function

Use when: the formula is built from pieces that each carry their own restriction — a square root, a fraction, a logarithm — and you need every input that survives all of them at once

Goal: the domain written as one interval or union of intervals, guaranteed to satisfy every restriction the formula imposes, simultaneously

Functions: domain, codomain, and composition

Write a direct proof of a universal claim

Use when: the claim has the form "for every $x$ in some set, $P(x)$ implies $Q(x)$ " (or just "for every $x$ , $Q(x)$ ") and a forward chain from hypothesis to conclusion is at least plausible

Goal: an unbroken chain of justified steps from an arbitrary element satisfying the hypothesis to the conclusion, valid for every such element at once

Reading and writing a proof

Prove a claim by contradiction

Use when: a direct forward chain from hypothesis to conclusion isn't obvious — especially when the claim asserts that something does NOT exist or CANNOT happen

Goal: a valid chain of steps from the claim's negation to an outright logical impossibility, which forces the original claim to be true

Reading and writing a proof

Vectors and geometry

Find the coefficient that produces a target vector

Use when: you're given vectors and a target vector, and asked to find a scalar coefficient $c$ so that a linear combination of the given vectors equals the target exactly

Goal: the coefficient value that makes the combination hit the target — or a clear demonstration that no such value exists

Linear combinations, first look

Test two vectors for orthogonality

Use when: you're given two vectors — possibly with an unknown entry — and asked whether they're orthogonal, or what value would make them orthogonal

Goal: a definite yes-or-no on orthogonality, or the exact unknown value that forces it, backed by a dot product that actually equals zero

Orthogonality

Compute a cross product in R3

Use when: you have two vectors in $\mathbb R^3$ and need a third vector perpendicular to both of them, or that vector's length for an area

Goal: the vector $\mathbf u\times\mathbf v$ , verified against both perpendicularity conditions, ready to use directly or to take the length of for an area

The cross product in R3

Find the vector and parametric equations of a line

Use when: you need a line's equation and you're given either a point with a direction vector, or two points known to lie on the line

Goal: a vector equation $\mathbf r(t)=\mathbf p_0+t\mathbf d$ , plus its coordinate-by-coordinate parametric equations, that generates every point on the line

Vector and parametric equations of a line

Classify a pair of lines in R3

Use when: you have two lines in $\mathbb{R}^3$ , each given as $\mathbf p_0 + t\mathbf d$ , and need to know whether they're parallel, intersecting, or skew

Goal: a definite classification — parallel (same or distinct), intersecting, or skew — plus the meeting point when one genuinely exists

Lines in R3, and skew lines

Uses the cross product above to test for coplanarity along the way.

Compute the scalar triple product

Use when: you have three vectors in $\mathbb{R}^3$ and need either the volume of the parallelepiped they span, or a yes/no test for whether the three are coplanar

Goal: the signed number $\det[\mathbf u\ \mathbf v\ \mathbf w]$ , read as a volume (its absolute value) or as a coplanarity test (zero or not)

Area, volume, and the scalar triple product

Solving a linear system

Solve a system of linear equations by Gaussian elimination

Use when: you have a system of linear equations and need its full solution, not just whether one exists

Goal: the numeric value of every unknown

Row echelon form and Gaussian elimination

Test existence and uniqueness from a matrix in row echelon form

Use when: you have (or can quickly reduce to) an augmented matrix in row echelon form, and need to know how many solutions the system has — not their actual values

Goal: a classification into exactly one of three outcomes: no solution, exactly one solution, or infinitely many

Existence and uniqueness

Reads the answer off an already-echelon matrix instead of solving it fully.

Detect that a system is inconsistent while eliminating

Use when: you're running Gaussian elimination on an augmented matrix and want to catch (or rule out) a contradiction as early as possible, rather than waiting until the whole matrix reaches echelon form

Goal: either a confirmed $0=c$ row — permission to stop immediately with no solution — or confidence that no such row exists anywhere, even after combining every row

Inconsistent systems, and what the picture looks like

Catches a contradiction mid-elimination rather than at the end.

Matrix algebra

Multiply two matrices

Use when: you have two matrices $A$ ( $n\times k$ ) and $B$ ( $k\times m$ ) and need the full product $AB$

Goal: the complete $n\times m$ product matrix, every entry computed and checked

Matrix multiplication

Compute a matrix power $A^k$

Use when: you have a square matrix $A$ and a positive integer $k$ , and need the matrix $A^k$

Goal: the matrix $A^k$ , found by whichever method is actually fastest for A's shape and k's size

Powers of a matrix

Multiply two block-partitioned matrices

Use when: you have two matrices split into blocks with matching block-column widths and block-row heights, and need one or more blocks of the product $C=AB$

Goal: the requested block (or blocks) of $C$ , computed by the same row-times-column rule as ordinary multiplication, applied one level up to sub-matrices

Block (partitioned) matrices

Working with a matrix's inverse and factorizations

Find a matrix inverse by row reduction

Use when: you have a square matrix $A$ and need $A^{-1}$ , for a size where no small shortcut formula applies

Goal: either the actual matrix $A^{-1}$ , or a confirmed conclusion that $A$ is singular and has none

Computing an inverse by row reduction

Solve Ax = b using a known inverse

Use when: you already have $A^{-1}$ in hand for an invertible matrix $A$ , and need to solve $A\mathbf x=\mathbf b$

Goal: the solution vector $\mathbf x$ , as one matrix-vector multiply instead of a fresh elimination

Solving Ax = b with an inverse -- and why you usually should not

Needs a known inverse already in hand -- see find-a-matrix-inverse above if you only have A.

Build an elementary matrix (and its inverse) for a row operation

Use when: you're told one specific row operation and asked for the matrix $E$ that performs it, or for $E^{-1}$ , without row-reducing anything

Goal: the matrix $E$ for that operation, checked by multiplying it into a real matrix, plus $E^{-1}$ if the problem asks for it

Elementary matrices

Factor a matrix into L and U

Use when: you have a square matrix $A$ and need $L$ (lower triangular, $1$ s on the diagonal) and $U$ (upper triangular) with $A=LU$

Goal: a complete $L$ and $U$ that multiply back out to $A$ exactly, with every multiplier ordinary elimination used recorded in $L$

LU factorization · extended later in this course

Factor a matrix into PA = LU (with pivoting)

Use when: you have a square matrix $A$ and ordinary elimination toward $A=LU$ hits a zero (or, for numerical work, an uncomfortably small) pivot

Goal: a permutation $P$ , unit lower-triangular $L$ , and upper-triangular $U$ with $PA=LU$

Permutation matrices, PA = LU, and pivoting

Extends the LU recipe above with the row-swap step ordinary elimination needs when a pivot is zero.

Determinants

Compute a 3×3 determinant by expanding along row 1

Use when: you have a $3\times3$ matrix and need its determinant, before the general cofactor-expansion technique is developed

Goal: the value of $\det A$ , plus what its sign and magnitude say about the parallelepiped $A$ 's columns span

The 3x3 determinant and volume

Solve a system by Cramer's rule

Use when: you have a square system $A\mathbf x=\mathbf b$ with $\det A\neq0$ and want one variable in isolation, as a ratio of two determinants, without eliminating the whole system

Goal: every $x_i$ , each computed as $\det A_i(\mathbf b)/\det A$ , with $A_i(\mathbf b)$ built by replacing exactly one column of $A$ with $\mathbf b$

Cramer's rule and the adjugate

Needs the determinant recipes above as its building block.

Vector spaces and subspaces

Test whether a set is a subspace

Use when: a problem hands you a subset $W$ of a vector space, defined by an equation, inequality, or other condition on its coordinates, and asks whether $W$ is a subspace

Goal: a proof that $W$ passes all three axioms for an arbitrary element and scalar, or one concrete counterexample showing exactly where it fails

Testing whether a set is a subspace

Find the null space of a matrix

Use when: you're given a matrix $A$ and asked for $\operatorname{Nul}(A)$ — every $\mathbf x$ with $A\mathbf x = \mathbf 0$ — rather than a single yes/no membership check

Goal: a spanning set for $\operatorname{Nul}(A)$ : one fixed vector per free variable, so every solution of $A\mathbf x=\mathbf 0$ is some combination of them

The null space · extended later in this course

Test whether a set of vectors is linearly independent

Use when: you're given a specific list of vectors $\mathbf v_1,\dots,\mathbf v_k$ (in any vector space, not necessarily $\mathbb{R}^n$ written as columns) and asked whether they're linearly independent

Goal: either a proof that only $c_1=\cdots=c_k=0$ solves the dependence equation, or an explicit nontrivial dependence relation showing which vector is redundant

Linear independence · extended later in this course

Test independence by row-reducing the coefficient matrix

Use when: the vectors live in $\mathbb{R}^n$ (or any setting where 'stack them as columns' makes sense) and you'd rather row-reduce a matrix than solve the dependence equation by hand substitution

Goal: a pivot-in-every-column verdict of independent, or a free column that hands you the dependence relation's coefficients directly

Independence, dependence, and homogeneous systems

The matrix/row-reduction version of the independence test above, for larger sets.

Find the dimension of a subspace

Use when: you're given some description of a subspace — a spanning set, a null space, a set defined by constraints — and asked how many vectors a basis for it needs

Goal: the exact number of vectors in any basis of the subspace, found without necessarily writing the basis out first

Dimension

Find a vector's coordinates relative to a basis

Use when: you have a basis $\mathcal B=\{\mathbf b_1,\dots,\mathbf b_n\}$ for a space $V$ and one specific vector $\mathbf x\in V$ , and need $[\mathbf x]_{\mathcal B}$

Goal: the unique list of scalars $c_1,\dots,c_n$ with $\mathbf x=c_1\mathbf b_1+\cdots+c_n\mathbf b_n$ , packaged as a column

Coordinates relative to a basis

The four fundamental subspaces

Find a basis for the column space

Use when: you have a specific matrix $A$ and need an actual, usable basis for $\operatorname{Col}(A)$ — not just its dimension

Goal: a linearly independent subset of $A$ 's own columns whose span is all of $\operatorname{Col}(A)$

Column space, revisited

Find the null space and verify its dimension

Use when: you're given a matrix $A$ , need $\operatorname{Nul}(A)$ itself, and want the pivot/free column count as a built-in check on the answer

Goal: a spanning set for $\operatorname{Nul}(A)$ together with confirmation that pivot columns plus free columns equals $A$ 's total column count

Null space, revisited

Extends the null-space recipe above with the rank-nullity dimension check.

Find the left null space of a matrix

Use when: you're given a matrix $A$ and asked for every row vector $\mathbf y$ with $\mathbf y^{\mathsf T}A=\mathbf 0^{\mathsf T}$ , rather than an ordinary (right) null space

Goal: a spanning set for the left null space, found by running the ordinary null-space recipe on $A^{\mathsf T}$ instead of $A$

The left null space

Extends the null-space recipe above, applied to $A^T$ .

Linear transformations and change of basis

Find the standard matrix of a linear transformation

Use when: you have a linear transformation's formula $T:\mathbb R^n\to\mathbb R^m$ and need the matrix $A$ with $T(\mathbf x)=A\mathbf x$

Goal: the $m\times n$ standard matrix $A$ , built column by column with nothing guessed

The standard matrix of a linear transformation

Find the matrix of a transformation relative to a basis

Use when: you have $T:\mathbb R^n\to\mathbb R^n$ , a basis $\mathcal B=\{\mathbf b_1,\dots,\mathbf b_n\}$ , and need the matrix $A$ with $A[\mathbf x]_{\mathcal B}=[T(\mathbf x)]_{\mathcal B}$

Goal: the matrix $A$ that runs $T$ entirely in $\mathcal B$ -coordinates, with no detour through standard coordinates

The matrix of a transformation relative to a basis

Compute (or confirm) a similarity $B = P^{-1}AP$

Use when: you have $A$ and an invertible $P$ , and either need $B=P^{-1}AP$ as a specific matrix, or need to confirm/deny a claimed $B$

Goal: either the actual matrix $B$ , or a settled yes/no on a claimed similarity — whichever the problem needs

Similar matrices

Orthogonality and least squares

Find the orthogonal complement of a subspace

Use when: you have a subspace $W$ given as a span of vectors and need $W^\perp$

Goal: $W^\perp$ described as the span of a basis, found by solving one homogeneous system

Orthogonal sets and orthogonal complements

Factor a matrix as $A=QR$

Use when: you have a matrix $A$ with linearly independent columns and need $Q$ (orthonormal columns) and $R$ (upper triangular) with $A=QR$

Goal: both $Q$ and $R$ , built together in one pass of Gram-Schmidt, with no separate computation for $R$ afterward

QR factorization

Solve a least-squares problem with the normal equations

Use when: you have an inconsistent system $A\mathbf x=\mathbf b$ (more conditions than the data can satisfy exactly) and need the best-fit $\hat{\mathbf x}$

Goal: the least-squares solution $\hat{\mathbf x}$ , found by solving one square system built from $A$ and $\mathbf b$

The normal equations

Solve a least-squares problem from an existing QR factorization

Use when: you already have $A=QR$ (orthonormal $Q$ , upper-triangular $R$ ) and need the least-squares solution to $A\mathbf x\approx\mathbf b$

Goal: $\hat{\mathbf x}$ , found with one matrix-vector product and one back substitution — no matrix ever multiplied by its own transpose

Least squares by QR

Skips straight to back-substitution when $A=QR$ is already in hand.

Working with one matrix's eigenstructure

Find the eigenvalues of a matrix

Use when: you have a square matrix $A$ and need the numbers $\lambda$ satisfying $A\mathbf v=\lambda\mathbf v$ for some nonzero $\mathbf v$

Goal: every eigenvalue of $A$ , as roots of one polynomial equation

Eigenvalues and the characteristic equation

Find or verify a $2\times2$ matrix's eigenvalues using trace and determinant

Use when: you have a $2\times2$ matrix — or a claimed pair of numbers said to be its eigenvalues — and want the eigenvalues confirmed without expanding $\det(A-\lambda I)$ by hand

Goal: the confirmed pair of eigenvalues, either derived directly or cleared against a candidate guess

Trace, determinant, and the eigenvalues

A faster path to the same eigenvalues above, for $2\times2$ matrices only.

Solve a characteristic equation with no real roots

Use when: you've formed $A-\lambda I$ for a $2\times2$ matrix and the resulting quadratic's discriminant, $(\operatorname{tr}A)^2-4\det A$ , comes out negative

Goal: the complex conjugate pair of eigenvalues, plus the correct conclusion that $A$ has no real eigenvector

When the characteristic polynomial has no real roots

What to do when the characteristic equation above has no real roots.

Find the eigenvectors for a given eigenvalue

Use when: you already have a matrix $A$ and one of its eigenvalues $\lambda$ , and need the eigenvectors that go with it

Goal: a basis for the eigenspace: every eigenvector belonging to $\lambda$ , described as a span

Eigenspaces

Needs an eigenvalue from the recipes above first.

Test whether a matrix is diagonalizable

Use when: you have a matrix's eigenvalues, at least one of which repeats, and need to know whether $n$ independent eigenvectors actually exist rather than just assuming they do

Goal: a confirmed yes-or-no on diagonalizability, with the exact eigenvalue and shortfall pinpointed if the answer is no

Similarity, diagonalizability, and defective matrices

Needed only when an eigenvalue repeats; checks whether enough independent eigenvectors actually exist.

Diagonalize a matrix

Use when: you're given a square matrix $A$ and asked to find $P$ and $D$ with $A=PDP^{-1}$ , $D$ diagonal

Goal: the eigenvalues plus enough independent eigenvectors to fill $D$ and $P$ , or a confirmed conclusion that $A$ can't be diagonalized

Diagonalization · extended later in this course

Needs eigenvalues and eigenvectors from above, then the diagonalizability check when a root repeats.

Compute $A^k$ by diagonalization

Use when: you have (or can find) $A=PDP^{-1}$ and need $A^k$ for some integer $k$ , rather than multiplying $A$ by itself $k$ times

Goal: $A^k$ , computed as $PD^kP^{-1}$ at a cost that doesn't grow with $k$

Powers of a matrix by diagonalization

Extends diagonalization above to compute $A^k$ without repeated multiplication.

Applications — Markov chains and differential equations

Find a Markov chain's exact steady state

Use when: you have a column-stochastic transition matrix $P$ and want the exact steady-state distribution $\mathbf x^\ast$ , not just its approximate behavior after many steps

Goal: the exact fractions making up $\mathbf x^\ast$ , satisfying both $P\mathbf x^\ast=\mathbf x^\ast$ and that its entries sum to 1

Markov chains and steady states

Solve $\dfrac{d\mathbf x}{dt}=A\mathbf x$ by eigenvalues and eigenvectors

Use when: you have a constant-coefficient linear system $\dfrac{d\mathbf x}{dt}=A\mathbf x$ (with or without an initial condition) and need its solution, or just the long-run behavior of its trajectories

Goal: the general solution $\mathbf x(t)$ , pinned to a specific trajectory if an initial condition is given, plus a stability classification of the origin

Linear systems of differential equations

Uses the eigenvalue and eigenvector recipes above to solve $\dfrac{d\mathbf x}{dt}=A\mathbf x$ .