Linear Algebra I

Year 1 · 7 sections · 98 concepts · 312 practice questions

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What's covered

free1 · Vectors and geometry13 concepts

Definition 1.1 · Vectors in ℝⁿ
Definition 1.2 · The dot product and length
Theorem 1.3 · The Cauchy–Schwarz inequality
Corollary 1.4 · The triangle inequality and angles
Definition 1.5 · Projection onto a vector
Proposition 1.6 · Projection gives the closest point
Definition 1.7 · The cross product in ℝ³
Proposition 1.8 · The scalar triple product
Definition 1.9 · Lines
Definition 1.10 · Planes in ℝ³
Proposition 1.11 · Distances to planes and lines
Example 1.12 · Intersections of lines and planes
Check 1.13 · Check: vector geometry problems

2 · Linear systems and matrices17 concepts

Definition 2.1 · Linear systems and augmented matrices
Proposition 2.2 · Row operations don't change the solutions
Theorem 2.3 · Row echelon form and reduced row echelon form
Theorem 2.4 · Reading off the solutions
Proposition 2.5 · Homogeneous systems with more unknowns than equations
Proposition 2.6 · General solution = particular + homogeneous
Definition 2.7 · Matrix operations
Proposition 2.8 · The algebra of matrices
Definition 2.9 · The transpose
Proposition 2.10 · Rules for the transpose
Definition 2.11 · Inverse matrices
Proposition 2.12 · Rules for inverses
Proposition 2.13 · Elementary matrices
Theorem 2.14 · The invertible matrix theorem
Technique 2.15 · Computing inverses by row reduction
Corollary 2.16 · For square matrices, a one-sided inverse is two-sided
Check 2.17 · Check: solving systems and inverting matrices

3 · Determinants12 concepts

Definition 3.1 · The determinant
Theorem 3.2 · The permutation (Leibniz) formula
Proposition 3.3 · Expansion along any row or column
Proposition 3.4 · Determinants of triangular matrices
Theorem 3.5 · Row operations and the determinant
Proposition 3.6 · det(Aᵀ) = det A
Theorem 3.7 · A is invertible iff det A ≠ 0
Theorem 3.8 · det(AB) = det A · det B
Theorem 3.9 · The adjugate and the inverse
Theorem 3.10 · Cramer's rule
Proposition 3.11 · Determinants as volume scale factors
Check 3.12 · Check: computing determinants

4 · Vector spaces15 concepts

Definition 4.1 · Vector spaces
Example 4.2 · Examples of vector spaces
Proposition 4.3 · Basic consequences of the axioms
Theorem 4.4 · Subspaces and the subspace test
Proposition 4.5 · Span
Definition 4.6 · Linear independence
Lemma 4.7 · The dependence lemma
Definition 4.8 · Bases and coordinates
Proposition 4.9 · Bases give unique coordinates
Theorem 4.10 · The Steinitz exchange lemma
Theorem 4.11 · Dimension
Proposition 4.12 · Extending and reducing to a basis
Proposition 4.13 · Dimensions of subspaces
Theorem 4.14 · Sums, direct sums and the dimension formula
Check 4.15 · Check: independence, spanning and bases

5 · Linear maps13 concepts

Definition 5.1 · Linear maps
Proposition 5.2 · A linear map is determined by a basis
Proposition 5.3 · Kernel and image
Theorem 5.4 · The rank–nullity theorem
Corollary 5.5 · Injective, surjective and bijective in equal dimensions
Definition 5.6 · The matrix of a linear map
Proposition 5.7 · Composition corresponds to matrix multiplication
Theorem 5.8 · Isomorphisms and dimension
Theorem 5.9 · Row rank equals column rank
Technique 5.10 · Bases for the null space and column space
Theorem 5.11 · Change of basis
Example 5.12 · Geometric linear maps of the plane
Check 5.13 · Check: working with linear maps

6 · Eigenvalues and diagonalisation14 concepts

Overview 6 · The plan of this section
Definition 6.1 · Eigenvalues and eigenvectors
Proposition 6.2 · The characteristic polynomial
Technique 6.3 · Finding eigenvectors and eigenspaces
Proposition 6.4 · Trace and determinant from eigenvalues
Proposition 6.5 · Similar matrices have the same eigenvalues
Theorem 6.6 · Eigenvectors for distinct eigenvalues are independent
Theorem 6.7 · Diagonalisable iff there is a basis of eigenvectors
Corollary 6.8 · n distinct eigenvalues ⇒ diagonalisable
Proposition 6.9 · Algebraic and geometric multiplicity
Proposition 6.10 · Powers of a diagonalisable matrix
Theorem 6.11 · The Cayley–Hamilton theorem
Example 6.12 · Complex eigenvalues
Check 6.13 · Check: diagonalising a matrix

7 · Inner products and orthogonality14 concepts

Overview 7 · The plan of this section
Definition 7.1 · Inner products
Proposition 7.2 · Cauchy–Schwarz and the triangle inequality in any inner product space
Proposition 7.3 · Orthogonal vectors are independent
Proposition 7.4 · Coordinates in an orthonormal basis
Theorem 7.5 · The Gram–Schmidt process
Theorem 7.6 · Orthogonal complements and projection
Theorem 7.7 · Projection is the best approximation
Example 7.8 · Least squares
Proposition 7.9 · Orthogonal matrices
Theorem 7.10 · Symmetric matrices have real eigenvalues and orthogonal eigenvectors
Theorem 7.11 · The spectral theorem for real symmetric matrices
Example 7.12 · Quadratic forms and conics
Check 7.13 · Check: orthogonality computations

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Theorem 1.3 · The Cauchy–Schwarz inequality

$|\mathbf{u}\cdot\mathbf{v}|\le\|\mathbf{u}\|\|\mathbf{v}\|$ , equality iff parallel. Proof: $\|\mathbf{u} - t\mathbf{v}\|^2\ge0$ for all $t$ ; discriminant $\le0$ . Makes angles possible: $\frac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}\in[-1, 1]$ .

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