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.

A polynomial or a complex inner product — readiness review: factor a polynomial and find each root's multiplicity, or compute $\langle u, v\rangle$ .

A general vector space fact — solve a linear equation over $\mathbb F_p$ , prove a set of functions is linearly independent, apply the subspace dimension formula, verify $V=U\oplus W$ , construct a complement and project onto it, or compute with cosets in $V/W$ .

A linear map or operator — verify an identity in $L(V,W)$ , build block-triangular form from an invariant subspace, evaluate a polynomial at an operator, or confirm nilpotency and find the smallest $k$ .

Duality — construct a dual basis, find a basis for an annihilator, apply the annihilator dimension formula, find the matrix of a dual map, or compute $T^*$ on a functional.

A bilinear, quadratic, or Hermitian form — evaluate a bilinear form, compute a new Gram matrix by congruence, classify a quadratic form by signature, or find an orthogonal complement under the form.

An inner product space — compute an inner product and check positive definiteness, find coordinates relative to an orthonormal basis, find the closest point in a subspace, or find the Riesz vector for a functional.

Spectral theory — find a complex eigenvalue and eigenvector, find the minimal polynomial, or build a spectral decomposition.

Canonical forms — determine diagonalizability via multiplicities, find the smallest $k$ for a generalized eigenvector, find a nilpotent operator's partition, decompose a vector via primary decomposition, or build the rational canonical form.

A function of a matrix, or a linear system's stability — evaluate a polynomial via Horner's method, compute a function of a diagonalizable matrix or a Jordan block, compute $e^{tA}$ , compute a matrix logarithm or square root, or classify a linear system from trace and determinant.

A pseudoinverse, or the SVD — compute a pseudoinverse via the SVD, find singular values by maximization, assess rank and conditioning, compute a truncated SVD's error, choose a rank for low-rank approximation, or run PCA via the SVD.

Spectral geometry or optimization — find a conic's principal axes, find an ellipsoid's semi-axes, test definiteness via leading principal minors, compute or bound a Rayleigh quotient, or solve a generalized eigenvalue problem.

Operators, norms, or perturbation — compute an operator norm, apply the Neumann series, or assess eigenvalue error from a residual.

Multilinear or exterior algebra — test whether a tensor is pure, classify and transform a tensor by type, compute a wedge product and confirm antisymmetry, compute an exterior power's dimension, work with the full Grassmann algebra, or determine orientation and unsigned volume.

Readiness

Factor a polynomial completely and state each root's multiplicity

Use when: you have a polynomial and need every root, over $\mathbb R$ or over $\mathbb C$ , along with how many times each one repeats

Goal: the polynomial written as a product of irreducible factors, with each root's multiplicity read straight off

Polynomials: division, roots, and factorization over ℝ and ℂ

Compute a complex inner product $\langle u, v\rangle$

Use when: you need $\langle u, v\rangle$ for vectors with complex-number entries, where the plain dot product no longer applies

Goal: the single complex number $\langle u,v\rangle = \sum u_k\overline{v_k}$ , with the conjugate on the correct side

Complex numbers and complex arithmetic you'll need

Vector spaces over a field

Solve a linear equation over $\mathbb F_p$

Use when: you have an equation like $ax+b=c$ with every number drawn from a finite field $\mathbb F_p$ , not from $\mathbb R$

Goal: the unique solution $x\in\{0,\dots,p-1\}$ satisfying the equation mod $p$

Vector spaces over ℚ, ℝ, ℂ, and 𝔽ₚ

Prove a set of functions is linearly independent

Use when: you have a finite list of functions and must show no nontrivial linear combination of them equals the zero function

Goal: a chain of forced equations $c_1=c_2=\dots=0$ , proving the only combination equal to zero is the trivial one

Infinite-dimensional spaces: polynomials, sequences, functions

Apply the subspace dimension formula

Use when: you know, or can find, three of $\dim U$ , $\dim W$ , $\dim(U+W)$ , $\dim(U\cap W)$ , and need the fourth

Goal: the missing dimension, found algebraically without necessarily constructing a basis for the harder subspace

Sums of subspaces and the dimension formula

Verify $V=U\oplus W$ and decompose a vector into its parts

Use when: you have two subspaces $U$ and $W$ and need to know whether $V=U\oplus W$ , or need a specific vector's unique $U$ -part and $W$ -part once it holds

Goal: either a confirmed direct sum with a genuine unique decomposition, or a concrete reason it fails

Direct sums

Construct a complement of $U$ and project a vector along it

Use when: you have a subspace $U$ and either need a valid complement $W$ , or need to project a specific vector onto $U$ along a given $W$

Goal: a verified complement $W$ , or a vector's unique split into its $U$ -part and its $W$ -part along that complement

Complements and the projections they define

A direct-sum decomposition where one summand is chosen deliberately, as a complement.

Compute with cosets in a quotient space $V/W$

Use when: you're working in $V/W$ and need to test coset equality, add or scale cosets, or find $\dim(V/W)$

Goal: the quotient space's dimension, or a specific coset correctly identified or combined with another

Quotient spaces V/W

Linear maps and the algebra of operators

Verify a claimed identity between maps in L(V,W)

Use when: you're given a candidate map built from other linear maps, such as a sum, a scalar multiple, or a claimed identity element, and need to confirm a stated equality actually holds

Goal: a vector-by-vector confirmation that the claimed equality holds for every vector in the domain, not just a few checked numbers

L(V,W) is itself a vector space

Build block-triangular form from one invariant subspace

Use when: you have one invariant subspace $U$ for an operator $T$ , with no complement required, and want a basis in which $T$ 's matrix has a forced zero block

Goal: a basis of $V$ and $T$ 's matrix relative to it, with the zero block explicitly identified and confirmed

Block-triangular form from an invariant subspace

Evaluate a polynomial at an operator or matrix

Use when: you're given a polynomial $p(x)=c_0+c_1x+\cdots+c_nx^n$ and an operator or matrix $A$ , and asked to compute $p(A)$

Goal: the single matrix (or operator) $p(A)$ , built entirely from powers of $A$ and the identity

The algebra L(V): composition, powers, polynomials in T

Confirm an operator is nilpotent and find the smallest $k$

Use when: you're given a matrix $N$ and need to determine whether some power $N^k$ equals the zero matrix, and if so, the smallest such $k$

Goal: either the smallest $k$ with $N^k=\mathbf 0$ , confirming nilpotency, or a specific power that stays nonzero, ruling nilpotency out

Nilpotent operators, first look

Duality

Construct a dual basis by solving directly

Use when: you have a specific basis $\{b_1,\dots,b_n\}$ of $V$ and need the functionals $\varphi_1,\dots,\varphi_n$ satisfying $\varphi_i(b_j)=\delta_{ij}$

Goal: every dual basis functional, each written as an explicit formula in coordinates

Dual bases

Find a basis for an annihilator $W^{\circ}$

Use when: you're given a spanning set for a subspace $W$ and need an explicit basis for its annihilator $W^{\circ}$

Goal: a set of explicit functionals that both annihilate $W$ and span all of $W^{\circ}$

Annihilators

Apply the annihilator dimension formula

Use when: you know two of $\dim V$ , $\dim W$ , $\dim(W^{\circ})$ and need the third, or want to relate $\dim(W^{\circ\circ})$ back to $\dim W$

Goal: the missing dimension, or the double-annihilator identity, without ever constructing a basis of $W$ or $W^{\circ}$

Annihilators and dimension

Solves the same identity a basis for the annihilator, from the recipe above, would let you find directly -- but needs no basis at all.

Find the matrix of a dual map

Use when: you know $T$ 's matrix relative to a basis and its dual basis, and need $T^{*}$ 's matrix or its exact effect on one given functional

Goal: the matrix of $T^{*}$ relative to the dual bases, and the explicit coordinates or formula for $T^{*}$ applied to any functional you're given

The matrix of the dual map is the transpose

Compute $T^{*}$ on a specific functional

Use when: you're given $T$ 's action in coordinates and a functional $\psi$ , and need $T^{*}(\psi)=\psi\circ T$ as an explicit formula

Goal: an explicit formula for $T^{*}(\psi)$ , as a functional of the original domain's coordinates

The dual (transpose) map T*

Bilinear, quadratic, and Hermitian forms

Evaluate a bilinear form and check its properties

Use when: you're given a two-argument formula and need its value at a specific pair, or need to confirm it's genuinely bilinear and check whether it's symmetric

Goal: the form's value at any given pair, plus confidence in whether it's linear in each argument and whether it's symmetric

Bilinear forms

Compute a new Gram matrix by congruence

Use when: you have a form's Gram matrix relative to one basis and a change-of-basis matrix $P$ , and need the Gram matrix relative to the new basis

Goal: the new Gram matrix $A'=P^{\mathsf T}AP$ , correctly distinguished from the similarity transform a linear map's matrix would use instead

Change of basis for forms: congruence, not similarity

Classify a real quadratic form by its signature

Use when: you have a real quadratic form, diagonal or not, and need to know how many positive, negative, and zero squares it reduces to, plus its standard classification

Goal: the signature $(p,q,r)$ and the matching name: positive definite, negative definite, indefinite, or degenerate

Signature and the classification of real quadratic forms

Often computed via the same congruence transformation as the recipe above.

Find the orthogonal complement of a subspace under a bilinear form

Use when: you have a bilinear form's Gram matrix and a subspace $W$ , and need $W^{\perp_B}=\{\mathbf v: B(\mathbf v,\mathbf w)=0 \text{ for every } \mathbf w\in W\}$

Goal: a spanning set for $W^{\perp_B}$ , with its dimension confirmed against $\dim V-\dim W$

Nondegenerate forms and orthogonal complements

Inner product spaces and adjoints

Compute an inner product and confirm positive definiteness

Use when: you're given vectors in $\mathbb R^n$ or $\mathbb C^n$ and need $\langle\mathbf u,\mathbf v\rangle$ computed correctly, or need to confirm a pairing is genuinely positive definite

Goal: the correct numeric value of $\langle\mathbf u,\mathbf v\rangle$ , together with confidence that pairing a vector with itself is real and positive

Inner products over R and C

Find coordinates relative to an orthonormal basis

Use when: you have a basis you know, or can check, is orthonormal, and need a vector's coordinates without solving a linear system

Goal: every coordinate $c_i=\langle\mathbf v,\mathbf e_i\rangle$ , confirmed by reconstructing $\mathbf v$ from them

Orthonormal bases and Fourier coefficients

Find the closest point in a subspace, and the minimum distance to it

Use when: the problem gives a subspace $W$ and a vector $\mathbf v$ , then asks for the closest point in $W$ , the minimum distance to it, or a proof that no point of $W$ beats a stated distance

Goal: the exact minimum distance from $\mathbf v$ to $W$ , backed by a proof no other point of $W$ can do better

Orthogonal projection and best approximation

Needs an orthonormal (or orthogonal) basis of the subspace, from the recipe above.

Find the Riesz vector for a linear functional

Use when: you have a linear functional $\varphi$ on a finite-dimensional inner product space and need the single vector $\mathbf w$ with $\varphi(\mathbf v)=\langle\mathbf v,\mathbf w\rangle$ for every $\mathbf v$

Goal: the unique vector $\mathbf w$ representing $\varphi$ , confirmed to reproduce $\varphi$ on every input

The Riesz representation theorem

Also built on an orthonormal basis, to write the functional's coordinates as a vector.

Spectral theory

Find an eigenvalue and eigenvector for a real matrix, over C

Use when: a real matrix's characteristic polynomial has a non-real root, or you're asked to work over $\mathbb C$ even when every root happens to be real

Goal: a complex eigenvalue $\lambda$ and a genuine eigenvector for it, verified by direct substitution

Every operator on a complex space has an eigenvalue

Find the minimal polynomial of a matrix

Use when: you have a square matrix $A$ and need the unique monic polynomial of smallest degree with $p(A)=0$

Goal: the minimal polynomial $m_A$ , confirmed to annihilate $A$ with no smaller degree working

The minimal polynomial

Build a spectral decomposition, and test simultaneous diagonalizability

Use when: you have a normal operator $T$ that needs to be written as $\sum_i\lambda_iP_i$ , or two such operators that need to be checked for a shared eigenbasis

Goal: the projections $P_i$ , the decomposition $T=\sum_i\lambda_iP_i$ , and a definite answer, when asked, on whether two operators share a common eigenbasis

Spectral decomposition as a sum of projections, and simultaneous diagonalization

Canonical forms

Determine whether a matrix is diagonalizable, via multiplicities

Use when: you have a square matrix's eigenvalues and need to know whether it's diagonalizable, without necessarily building $P$ and $D$

Goal: a definite conclusion, diagonalizable or not, backed by an exact multiplicity comparison at every eigenvalue

Why diagonalization can fail

Find the smallest k for a generalized eigenvector

Use when: you have a vector $\mathbf v$ and an eigenvalue $\lambda$ , and need to know whether $\mathbf v$ is a generalized eigenvector, or the smallest $k$ with $(T-\lambda I)^k\mathbf v=\mathbf 0$

Goal: the smallest exponent $k$ , plus the ordinary eigenvector that shows up one application before it

Generalized eigenvectors

Find the partition of a nilpotent operator

Use when: you have a nilpotent operator or matrix $N$ and need the exact sizes of its Jordan chains, the data that fixes its similarity class completely

Goal: the full partition $(k_1,k_2,\dots)$ of chain lengths, with the lengths summing to $\dim V$

Nilpotent operators and their structure

Decompose a vector using primary decomposition

Use when: you know $T$ 's minimal polynomial factored by eigenvalue and need to split a vector (or all of $V$ ) into its generalized-eigenspace pieces

Goal: one projection operator per eigenvalue, and the exact generalized-eigenspace decomposition of any vector in $V$

Primary decomposition

Build the rational canonical form from the characteristic and minimal polynomials

Use when: you know $T$ 's characteristic polynomial $p_T$ and minimal polynomial $m_T$ and need $T$ 's invariant factors, elementary divisors, or its rational canonical form

Goal: the full list of invariant factors, assembled into companion-matrix blocks that build the rational canonical form

Rational canonical form, invariant factors, and elementary divisors

Draws on the minimal polynomial and the nilpotent partition, from earlier recipes.

Polynomial and functional calculus

Evaluate a polynomial at a matrix using Horner's method

Use when: you need $p(T)$ for a specific polynomial $p$ and operator or matrix $T$ , and want fewer matrix multiplications than computing every power separately

Goal: the matrix $p(T)$ , computed with exactly $n$ matrix multiplications for a degree- $n$ polynomial

Polynomials of an operator, revisited

Compute a function of a diagonalizable matrix

Use when: you have a diagonalizable matrix $A$ and a function $f$ defined at each of $A$ 's eigenvalues, not necessarily a polynomial

Goal: the matrix $f(A)$ , built by applying $f$ to $A$ 's eigenvalues and conjugating back

Functions of a diagonalizable matrix

Compute a function of a Jordan block

Use when: you need $f(A)$ for an operator $A$ that isn't diagonalizable, given $A$ 's Jordan form, and $f$ has enough derivatives at each eigenvalue

Goal: $f(J)$ for each Jordan block, assembled and conjugated back into $f(A)$

Functions of a general matrix, via Jordan blocks

The non-diagonalizable counterpart to the recipe above.

Compute $e^{tA}$ using the Jordan-Chevalley decomposition

Use when: you need $e^{tA}$ for an operator $A$ and have (or can find) its Jordan-Chevalley decomposition $A=S+N$ , with $S,N$ commuting, $S$ diagonalizable and $N$ nilpotent

Goal: the closed-form matrix $e^{tA}=e^{tS}e^{tN}$ , with each factor computed by an already-known, easier method

The matrix exponential

A specific function (the exponential) via the same Jordan-block machinery.

Compute a matrix logarithm or square root

Use when: you need $\log A$ or $\sqrt A$ for a diagonalizable matrix $A$

Goal: the matrix $\log A$ or $\sqrt A$ , correctly flagged as real, complex, or undefined based on $A$ 's eigenvalues

Matrix logarithms and square roots

Uses the same diagonalizable-function machinery, run in reverse.

Classify a linear system's phase portrait from trace and determinant

Use when: you have $\mathbf x'=A\mathbf x$ for a $2\times2$ real matrix $A$ and want the portrait type and stability without solving for eigenvalues

Goal: the phase portrait type (node, saddle, spiral, or center) and whether the system is stable, computed from $\text{tr}(A)$ and $\det(A)$ alone

Linear systems x' = Ax, stability, and phase portraits

Matrix factorizations

Compute a matrix's pseudoinverse

Use when: you need $A^{+}$ for a matrix $A$ that may be singular, rectangular, or otherwise not invertible

Goal: the pseudoinverse $A^{+}=V\Sigma^{+}U^{\mathsf T}$ , matching $A^{-1}$ exactly whenever $A$ genuinely is invertible

The pseudoinverse

The singular value decomposition

Find singular values and vectors from the maximum-stretch definition

Use when: $A$ is diagonal (or already diagonalized) and you want $\sigma_i$ , $\mathbf v_i$ directly from the variational definition, not from $A^{\mathsf T}A$

Goal: every singular value and its matching right singular vector, found one maximization at a time

Singular values and singular vectors

Assess a matrix's rank and conditioning from its singular values

Use when: you have $A$ 's singular values (or its SVD) and need its rank, how trustworthy that rank is numerically, or how sensitive $A\mathbf x=\mathbf b$ is to input error

Goal: the exact rank, the numerical rank at a chosen tolerance, and the condition number $\kappa(A)$ , with what each one means for solving with $A$

Rank, numerical rank, and the condition number

Compute a truncated SVD's approximation error

Use when: you truncate $A$ 's SVD to rank $k$ and need the error $\lVert A-A_k\rVert$ , or want to know if a better rank- $k$ matrix could exist

Goal: the operator-norm and Frobenius-norm error of the rank- $k$ truncation, both read directly off the discarded singular values

The Eckart-Young theorem

Choose a rank for low-rank approximation

Use when: you know $A$ 's singular values (or its dimensions) and need either the storage cost at a chosen rank $k$ , or the smallest $k$ meeting a target relative error

Goal: a rank $k$ (or the cost and error that rank produces), decided from the singular value spectrum alone, with no approximation ever built first

Low-rank approximation in practice

Uses the truncation-error formula from the recipe above, run in reverse.

Run PCA via the SVD directly

Use when: you need principal components or explained variance for a centered data matrix $X$ , without ever forming the covariance matrix

Goal: the principal directions and the variance each one explains, computed entirely from $X$ 's own singular values

PCA is an SVD

Spectral geometry and optimization

Find the principal axes and standard form of a conic or quadric

Use when: you have a two- or three-variable quadratic expression with a nonzero cross term ( $xy$ , $xz$ , or $yz$ ) and need its axes or its cross-term-free standard form

Goal: the orthonormal eigenbasis that removes the cross term, plus the standard form in that basis

Conics, quadrics, and the principal-axis theorem

Find an ellipsoid's semi-axes and read its shape from a positive definite matrix

Use when: you have a positive definite matrix $A$ and need the semi-axis lengths, axis directions, or overall shape of the level set $\{x^{\mathsf T}Ax=1\}$

Goal: every semi-axis length, its matching direction, and a shape comparison based on how spread out the eigenvalues are

Ellipsoids and the shape of a positive definite matrix

Test definiteness using leading principal minors

Use when: you have a small symmetric matrix given as numbers, not eigenvalues, and want a definiteness verdict without computing a full eigendecomposition

Goal: a positive-definite, negative-definite, or 'neither' verdict, certified by a short list of ordinary determinants

Four tests for definiteness, and when each is the right one

Compute the Rayleigh quotient at a vector

Use when: you have a symmetric matrix $A$ and a specific nonzero vector $x$ , and need the single number $R_A(x)$

Goal: the value of $R_A(x)$ , plus whether it equals an eigenvalue exactly

The Rayleigh quotient

Bound or confirm the extreme eigenvalues using the Rayleigh quotient

Use when: you want to estimate, bound, or confirm $\lambda_{\max}$ or $\lambda_{\min}$ of a symmetric matrix without first solving the full characteristic equation

Goal: a certified lower bound on $\lambda_{\max}$ (or upper bound on $\lambda_{\min}$ ), tightened to an exact value once a genuine eigenvector turns up

Rayleigh-Ritz and the extreme eigenvalues

Needs the Rayleigh quotient itself, from the recipe above.

Solve a generalized eigenvalue problem

Use when: you're asked to maximize or minimize $x^{\mathsf T}Ax$ subject to $x^{\mathsf T}Bx=1$ , or directly to solve $Ax=\lambda Bx$ , with $B$ symmetric positive definite

Goal: every generalized eigenvalue and its matching eigenvector, normalized against $B$ rather than against the identity

Constrained quadratic optimization and the generalized eigenvalue problem

A B-weighted version of the same extreme-eigenvalue bounding above.

Operators, norms, and infinite dimensions

Compute the operator norm of a diagonal or symmetric matrix

Use when: you need $\lVert A\rVert$ for a diagonal matrix, or a symmetric matrix whose eigenvalues you can find

Goal: the exact value of $\lVert A\rVert$ , found directly from the definition rather than from a formula not introduced yet

Matrix norms and operator norms

Apply the Neumann series to invert $I-A$ or bound the resolvent

Use when: you need $(I-A)^{-1}$ or $(A-\lambda I)^{-1}$ written as an infinite matrix power series, or need to know exactly where such a series is valid

Goal: either the literal sum, when it terminates or has a recognizable form, or the precise region where the series is guaranteed to converge

The Neumann series and the resolvent

Assess how much a residual guarantees about an eigenvalue

Use when: you have a candidate eigenpair $(x,\lambda)$ with residual $r=Ax-\lambda x$ and want to know how far $\lambda$ actually sits from a genuine eigenvalue

Goal: a residual-based bound on the error when $A$ is normal, or an honest admission that no such bound exists when $A$ is not

Approximate eigenvectors and why the spectrum can be unstable

Multilinear and exterior algebra

Test whether a tensor is pure

Use when: you have a specific element of $V\otimes W$ written as a sum of basis tensors $e_i\otimes f_j$ and need to know whether it equals a single $v\otimes w$

Goal: either an explicit $v$ and $w$ decomposing the tensor, or a proof that no such $v,w$ exist

The tensor product and its universal property

Classify a tensor's type and transform its components

Use when: you're given a tensor, or its transformation rule under a basis change $P$ , and need its $(p,q)$ type or its correctly transformed components

Goal: the tensor's $(p,q)$ type, and its components transformed correctly using $P^{-1}$ for each contravariant factor and $P^{\mathsf T}$ for each covariant factor

Covariant, contravariant, and mixed tensors

Compute u wedge v and confirm the sign flip under a swap

Use when: you have two vectors $u,v \in \mathbb R^2$ and need the signed number $u\wedge v$ , or want to verify wedge's alternating property directly

Goal: the signed area $u\wedge v = u_1v_2-u_2v_1$ , with the swap $v\wedge u=-(u\wedge v)$ confirmed rather than assumed

The wedge product

Compute the dimension of an exterior power

Use when: you're given $\dim V=n$ and an exponent $k$ , and need $\dim(\Lambda^kV)$

Goal: the exact dimension of $\Lambda^kV$ , correctly zero when $k$ exceeds $n$ and a binomial coefficient otherwise

Exterior powers Λᵏ V and the dimension count

Work with the full Grassmann algebra $\Lambda V$

Use when: you're asked either for $\dim(\Lambda V)$ , the total dimension of the full exterior algebra, or to say what a differential $k$ -form adds beyond this chapter's own $\Lambda^k(V^{*})$

Goal: the numeric total $2^n$ , or a one-line statement of the smooth-variation ingredient a differential form adds

The Grassmann algebra and where differential forms pick up

Combines every exterior power from the recipe above into one structure, and names the boundary with a later course's differential forms.

Determine orientation and unsigned volume from one determinant

Use when: you have an ordered list of $n$ vectors in $\mathbb R^n$ and need both their orientation and their unsigned volume

Goal: the vectors' orientation (positive, negative, or undefined) and their unsigned volume, both read from one determinant

Oriented area, volume, and orientation