Linear Algebra · Lesson 04

Course review

The whole course in one sitting: systems and matrices, determinants and inverses, elimination, vector spaces and linear maps, eigenvalues, orthogonality and least squares, and the spectral theorem. Each question needs a different chapter, and no worked example sits above the answers.

Foundations8 min readRevision 1

Read freely. Sign in when you want to save your place.

Sign in to save progress

A x = b

A v = λ v

det (A)

u \cdot v

(a c b d)

Chapter navigation

Intuition

The whole course in one sitting: systems and matrices, determinants and inverses, elimination, vector spaces and linear maps, eigenvalues, orthogonality and least squares, and the spectral theorem. Each question needs a different chapter, and no worked example sits above the answers.

At the end of a journey the map makes sense in a way it could not at the start: every road leads somewhere already visited. These questions walk the roads once more.

The thread of the course

One object runs through everything: the linear combination. A system asks which combination of columns gives $\mathbf{b}$ ; a matrix is a map built from combinations; a span, a basis and a dimension describe the combinations of a list; an eigenvector is a direction a map only scales; a projection is the combination closest to a given vector.

c_1\mathbf{v}_1+c_2\mathbf{v}_2+\cdots+c_n\mathbf{v}_n

Reminders

$A\mathbf{x}=\mathbf{b}$ is solvable exactly when $\operatorname{rank}A=\operatorname{rank}[A\mid\mathbf{b}]$ .
A square matrix is invertible exactly when $\det A\neq0$ , when it has $n$ pivots, when $0$ is not an eigenvalue.
$\dim V=\dim\ker T+\dim\operatorname{im}T$ .
$A=PDP^{-1}$ with a basis of eigenvectors; $A=QDQ^{T}$ when is symmetric.
$A^{T}A\hat{\mathbf{x}}=A^{T}\mathbf{b}$ for least squares.

Applications