Gaussian Elimination and Linear Systems

1.1 Linear Systems and their Solutions

1.1.1 General Equation of a Line

• Linear equation in xy-plane: ax + by = c

• Gradient: -a/b, y-intercept: c/b

1.1.2 Extension to n Variables

• General form: a1x1 + a2x2 + ... + anxn = b

1.1.3 Geometric Interpretation in R2 and R3

• Types of intersections between lines: parallel with no solutions, coincident with infinitely many solutions, and intersecting with one solution.

• Planes in R3 can also have similar intersections:

  1. Case of no solutions: Parallel planes

  2. Infinitely many solutions: Coincident planes

  3. One solution: Intersecting at a line

1.2 Elementary Row Operations (EROs)

• Types of EROs:

  1. Multiply an equation by a non-zero constant

  2. Interchange two equations

  3. Add a multiple of one equation to another

1.2.1 Row Equivalence

• Two matrices are row equivalent if one can be obtained from another through a series of EROs.

1.2.2 Gaussian Elimination and REF

• REF characteristics:

  1. All-zero rows at the bottom

  2. Leading entry of a row is to the right of the leading entry of the previous row.

1.3 Applications

1.3.1 Macroeconomics: Input-Output Analysis

• Developed by Wassily Leontief, shows the inter-industry relationships in an economy and how changes in one sector affect others.

1.3.2 Electrical Circuits: Kirchhoff’s Laws

• Kirchhoff’s Current Law: sum of currents at a junction is zero.

• Kirchhoff’s Voltage Law: sum of potential differences around a loop is zero.

1.3.3 Recreational Mathematics: Magic Square

• A magic square is an arrangement where sums of each row, column, and diagonal are equal.

Matrices

2.1 Matrix Operations

2.1.1 Matrix Multiplication

• Derived from element-wise multiplication of rows and columns.

2.1.2 Laws of Matrix Operations

• Commutative and associative laws for addition, distributive laws for multiplication.

2.5 Inverse of Square Matrices

• A square matrix A is invertible if there exists a matrix B such that AB = I.

• If A is singular, then det(A) = 0.

2.6 Determinants

2.6.1 Influence of EROs on Determinant

• EROs affect determinant:

  1. Multiplying a row by a constant scales the determinant.

  2. Interchanging two rows negates the determinant.

2.6.2 Rule of Sarrus

• A method for finding determinants of 3x3 matrices easily.

Vector Spaces

3.1 Euclidean n-Space

• Definition of n-vectors and properties of addition, scalar multiplication, etc.

3.3 Linear Combination and Span

3.3.1 Linear Combination

• Definition and examples of forming combinations of vectors.

3.3.2 Standard Basis

• Formed by unit vectors in the coordinate directions.

3.4 Subspaces

• Definition of subspaces in relation to R^n.

• Criteria for subset to be a subspace.

3.6 Basis and Dimension

3.6.1 Basis

• A basis must meet independence and span conditions.

3.6.2 Basis Theorem

• States conditions for linear independence and spanning a vector space.

Orthogonality

5.1 Dot Product and Lp Norm

• Length calculation and definitions of norms defined on vectors.

5.2 Orthogonal and Orthonormal Bases

• Definitions and examples of basis sets.

Linear Transformations

7.1 Linear Transformations from Rn to Rm

• Definition and standard matrix representation.

7.3 Kernel and Nullity

7.3.1 Invertible Matrix Theorem

• Summarizes several properties and criteria for matrices associated with linear transformations.

7.4 Geometric Transformations in R2 and R3

• Types of transformations: Translation, Scaling, Reflection, Shear, Rotation.