Chapter 1: Vectors and Matrices

Strang, Introduction to Linear Algebra (6th Edition)

Summary

Chapter 1 builds the language the rest of the book uses. Vectors combine into linear combinations, dot products measure lengths and angles, and a matrix holds vectors as its columns. The column space of a matrix is every combination of those columns, and multiplying two matrices composes the transformations they describe. The chapter ends by factoring a matrix as \(A = CR\), its independent columns times the recipe that rebuilds the rest.

flowchart LR
  V[Vectors] --> LC[Linear Combinations]
  LC --> DP[Dot Products, Lengths and Angles]
  LC --> CS[Matrices and Their Column Spaces]
  CS --> MM[Matrix Multiplication]
  MM --> CR[A = CR]

Sections

Section	Topic	Status
1.1	Vectors and Linear Combinations	Not Started
1.2	Lengths and Angles from Dot Products	Not Started
1.3	Matrices and Their Column Spaces	Not Started
1.4	Matrix Multiplication AB and CR	Draft

Review Questions

Question 1 Write \((3, 5)\) as a combination \(c(1, 1) + d(1, 2)\).

Answer

\(c = 1\) and \(d = 2\), since \((1, 1) + 2(1, 2) = (3, 5)\).

Question 2 Find the angle between \((1, 0)\) and \((1, 1)\).

Answer

\(\cos\theta = \dfrac{1}{1 \cdot \sqrt{2}}\), so \(\theta = 45^\circ\).

Question 3 For a 3 by 2 matrix \(A\) and a 2 by 3 matrix \(B\), give the sizes of \(AB\) and \(BA\).

Answer

\(AB\) is 3 by 3 and \(BA\) is 2 by 2, so both products exist but they cannot be equal.

Formula Bank

Rule	Statement	Holds when
Size	\((m \times n)(n \times p)\) gives \(m \times p\)	The inner sizes agree
Entry	\((AB)_{ij} = \sum_k a_{ik} b_{kj}\)	Always
Columns	Column \(j\) of \(AB\) is \(A\) times column \(j\) of \(B\)	Always
Rows	Row \(i\) of \(AB\) is row \(i\) of \(A\) times \(B\)	Always
Associative	\((AB)C = A(BC)\)	Always
Transpose	\((AB)^{\mathsf T} = B^{\mathsf T} A^{\mathsf T}\)	Always
Order	\(AB = BA\)	Only for special pairs, not in general
Zero product	\(AB = 0\) forces \(A = 0\) or \(B = 0\)	Never in general