Linear Algebra II

Year 2 · 7 sections · 71 concepts · 181 practice questions

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What's covered

free1 · Quotients and direct sums11 concepts

Overview 1 · The plan of this section
Definition 1.1 · Direct sum
Proposition 1.2 · Tests for a direct sum
Proposition 1.3 · Every subspace has a complement
Definition 1.4 · Projection
Proposition 1.5 · Projections and direct sums
Definition 1.6 · Quotient space
Proposition 1.7 · The quotient is a vector space of dimension dim V − dim U
Theorem 1.8 · First isomorphism theorem
Proposition 1.9 · Induced operators on subspaces and quotients
Check 1.10 · Check: direct sums and quotients

2 · Polynomials of operators11 concepts

Overview 2 · The plan of this section
Definition 2.1 · Polynomials in an operator
Definition 2.2 · Invariant subspace
Proposition 2.3 · Kernels of polynomials are invariant
Definition 2.4 · Minimal polynomial
Proposition 2.5 · The minimal polynomial divides every annihilating polynomial
Proposition 2.6 · Roots of the minimal polynomial are the eigenvalues
Theorem 2.7 · Cayley–Hamilton theorem
Theorem 2.8 · Diagonalisable iff the minimal polynomial has distinct linear factors
Theorem 2.9 · Primary decomposition
Check 2.10 · Check: computing the minimal polynomial

3 · Jordan normal form11 concepts

Overview 3 · The plan of this section
Definition 3.1 · Generalised eigenspace
Proposition 3.2 · V is the direct sum of generalised eigenspaces
Definition 3.3 · Nilpotent operators and Jordan blocks
Lemma 3.4 · Jordan chains are linearly independent
Theorem 3.5 · Structure of nilpotent operators
Proposition 3.6 · Counting Jordan blocks with kernel dimensions
Theorem 3.7 · Jordan normal form
Example 3.8 · Computing a Jordan basis
Proposition 3.9 · Powers and exponentials of Jordan blocks
Check 3.10 · Check: Jordan form computations

4 · Dual spaces10 concepts

Overview 4 · The plan of this section
Definition 4.1 · Dual space and dual basis
Proposition 4.2 · The dual basis is a basis
Proposition 4.3 · The double dual
Definition 4.4 · Annihilator
Proposition 4.5 · Dimension of the annihilator
Definition 4.6 · Dual map
Proposition 4.7 · The dual map has the transpose matrix
Corollary 4.8 · Row rank equals column rank
Check 4.9 · Check: dual basis computations

5 · Inner product spaces9 concepts

Overview 5 · The plan of this section
Definition 5.1 · Real and complex inner products
Theorem 5.2 · Cauchy–Schwarz and the triangle inequality
Proposition 5.3 · Orthonormal bases
Theorem 5.4 · Orthogonal complement and orthogonal projection
Theorem 5.5 · Riesz representation
Definition 5.6 · Adjoint
Proposition 5.7 · Properties of the adjoint
Check 5.8 · Check: complex inner product computations

6 · Spectral theorems10 concepts

Overview 6 · The plan of this section
Definition 6.1 · Self-adjoint, normal and unitary operators
Proposition 6.2 · Eigenvalues and eigenvectors of special operators
Theorem 6.3 · Schur's theorem
Theorem 6.4 · Complex spectral theorem
Theorem 6.5 · Real spectral theorem
Proposition 6.6 · Unitary operators preserve the inner product
Definition 6.7 · Positive operators
Theorem 6.8 · Singular value decomposition
Check 6.9 · Check: orthogonal diagonalisation

7 · Bilinear and quadratic forms9 concepts

Overview 7 · The plan of this section
Definition 7.1 · Bilinear forms and their matrices
Proposition 7.2 · Change of basis for forms
Definition 7.3 · Quadratic forms and polarisation
Theorem 7.4 · Symmetric forms diagonalise
Theorem 7.5 · Sylvester's law of inertia
Proposition 7.6 · Tests for definiteness
Example 7.7 · Classifying conics and quadrics
Check 7.8 · Check: quadratic forms

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Proposition 1.2 · Tests for a direct sum

Direct ⇔ only trivial way to write $0$ ⇔ dimensions add ⇔ bases concatenate. Two subspaces: ⇔ $U_1\cap U_2 = 0$ . Three or more: pairwise zero intersections are not enough.

Proof

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