Sets

\(\mathbb{R}\) is the set of real numbers (rational and irrational)

\(x \in \mathbb{R}\) denotes that \(x\) belongs to the set of real numbers

Consider the universe of all real numbers, with set \(A\) given by \[A = \{x \mid x \geq 2\}\]

What is the complement of \(A\)?

Set Relations and Operations

Consider the following sets: \[\begin{aligned} A &= \{x \mid x \geq 1\} \\ B &= \{x \mid x \text{ is a positive integer}\} \\ C &= \{x \mid 2 < x < 7\} \end{aligned}\]

• Is \(A=B\)? Is \(B \subset A\)?

• Are \(C\) and \(B\) disjoint sets?

• What is \(A \cap B\)? What about \(A \cup B\)?

• What is \(B \cap C\)?

Subsets

\(\emptyset\): empty or null set

What are all possible subsets of \[S = \{a,b,c\}\]

\(\emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{b,c\}, \{a,c\}, \{a,b,c\}\)

Always \(2^n\) subsets. Here \(n=3\), so 8 subsets.

Set Operations: Venn Diagrams

Shade the following regions:

Cartesian Product

The Cartesian product of two sets \(A\) and \(B\) is the set of all ordered pairs whose first element comes from \(A\) and second from \(B\): \[A \times B = \{(a,b) \mid a \in A, \; b \in B\}\]

Example. \(A = \{0,1\}\), \(B = \{5,6,7\}\). What is \(A \times B\)? How many pairs?

Cartesian Plane

\[\mathbb{R}^2 = \{(x,y) \mid x \in \mathbb{R}, y \in \mathbb{R}\}\]

Can have \(\mathbb{R}^3, \mathbb{R}^4,...,\mathbb{R}^n\)

Relations and Functions

Relation: subset of the Cartesian product, e.g. \(\{(x,y) \mid y \leq x \}\)

Function: a relation where for each \(x\) there is a unique \(y\) \[f: X \rightarrow Y, \quad y = f(x)\]

\(X:\) domain, \(Y:\) codomain, \(f(X):\) range

Is \(y = |x|\) a function? Is \(\{(x,y) \mid x = y^2\}\) a function of \(x\)?

Inverse of a Function

Function \(y=f(x)\) has an inverse if it is a one-to-one mapping, i.e. each value of \(y\) is associated with a unique value of \(x\).

The inverse function \[x=f^{-1}(y)\] returns the corresponding value of \(x\) for each \(y\).

One-to-one mapping is unique to strictly monotonic functions.

Find the inverse of \(y = 4x + 8\). Does \(y = x^2\) have an inverse?

Summation Notation

\[\sum_{i=1}^N x_i = x_1 + x_2 + ... + x_N\]

Let \(x_1 = 3\), \(x_2 = 5\), \(x_3 = 1\), \(x_4 = 7\).

• Evaluate \(\sum_{i=2}^{4} x_i\)

• Evaluate \(\sum_{i=1}^{4} (2 x_i + 3)\)

• Expand \(\sum_{i=1}^{3} b_i x^{i}\)

Necessary vs. Sufficient Conditions

\(q\) is a necessary condition for \(p\) if \(p \implies q\)

\(q\) is a sufficient condition for \(p\) if \(q \implies p\)

If both hold, \(p \iff q\) and each is necessary and sufficient for the other

Example. \(p\): you got an A in this course; \(q\): you passed this course.

Is passing necessary for an A? Is it sufficient?

Matrices

A matrix with \(m\) rows and \(n\) columns is an \(m \times n\) matrix \[A = [ a_{ij} ] \quad i=1,2,...,m; \; j=1,2,...,n\]

Square matrix: equal number of rows and columns

Two matrices are equal if all their elements are identical

What is the dimension of \(A = \begin{bmatrix} 5 & 0 \\ 2 & 7 \\ 1 & 3 \end{bmatrix}\)?

Addition and Scalar Multiplication

Element-by-element, and the matrices must have the same dimension: \[A = \begin{bmatrix} 1 & 4 \\ 3 & 2 \end{bmatrix} \quad B = \begin{bmatrix} 5 & 0 \\ -1 & 6 \end{bmatrix}\]

Find \(A + B\), \(A - B\), and \(3A\).

Matrix Multiplication

Only possible to multiply \(A_{m \times n}\) and \(B_{p \times q}\) to get \(AB\) if \(n = p\): \[\text{number of columns in } A = \text{ number of rows in } B\]

\[C = AB, \qquad c_{ij} = a_{i1} b_{1j} + a_{i2} b_{2j} + ... + a_{in} b_{nj} = \sum_{k=1}^n a_{ik} b_{kj}\]

The element \(c_{ij}\) comes from the \(i\)th row of \(A\) and the \(j\)th column of \(B\).

Matrix Multiplication

\[A = \begin{bmatrix} 1 & 0 & 2 \\ 3 & 1 & -1 \end{bmatrix}_{2 \times 3} \quad B = \begin{bmatrix} 2 \\ 4 \\ 1 \end{bmatrix}_{3 \times 1}\]

Is \(AB\) defined? What is its dimension? Find it.

What about \(BA\)?

Vectors and Linear Dependence

Matrices with only one column are column vectors; with only one row, row vectors

A set of vectors is linearly dependent if and only if any one of them can be expressed as a linear combination of the remaining vectors

Are these linearly independent? \[v_1 = \begin{bmatrix} 2 \\ 6 \end{bmatrix} \quad v_2 = \begin{bmatrix} 1 \\ 3 \end{bmatrix} \qquad \qquad u_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \quad u_2 = \begin{bmatrix} 2 \\ 5 \end{bmatrix}\]

Special Matrices

Identity matrix \(I\): \(1\)s on the principal diagonal, \(0\)s elsewhere; acts like 1, so \(AI = IA = A\)

Null matrix: all elements \(0\); \(A + 0 = A\) and \(A0 = 0\)

Idempotent: \(A\) is idempotent if \(A = A^k\), i.e. it is unchanged when multiplied by itself

Is an identity matrix idempotent?

Transpose of a Matrix

\(A^T\): interchange rows and columns

Find \(A^T\) for \[A = \begin{bmatrix} 1 & 0 & 4 \\ 2 & 5 & 3 \end{bmatrix}\]

What is the dimension of \(A^T\)? What about \(A^T A\)?

Inverse of a Matrix

For a square matrix \(A\), its inverse \(A^{-1}\) is defined as \[A A^{-1}=A^{-1} A=I\]

If the inverse exists, \(A\) is nonsingular

Squareness is necessary but not sufficient

Conditions for Nonsingularity

Sufficient condition: rows (or equivalently) columns are linearly independent

\[A=\begin{bmatrix} 2 & 6 \\ 1 & 3 \end{bmatrix} \quad B=\begin{bmatrix} 2 & 6 \\ 1 & 4 \end{bmatrix}\]

Which is singular, and which is nonsingular?

Determinant

The determinant \(|A|\) is a unique scalar associated with a square matrix \(A\)

For a \(2 \times 2\) matrix: \[|A|=a_{11} a_{22}-a_{12} a_{21}\]

A minor \(|M_{ij}|\) deletes the \(i\)th row and \(j\)th column; the cofactor is \[|C_{ij}| = (-1)^{i+j} |M_{ij}|\]

Then, expanding along any row or column: \[|A| = \sum_{j=1}^n a_{ij} |C_{ij}|\]

Find the Determinant

\[A=\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}\]

Is \(A\) nonsingular?

Criteria for Nonsingularity

The following statements are equivalent:

• \(|A| \neq 0\)

• Rows (or equivalently columns) of \(A\) are independent

• \(A\) is nonsingular

• \(A^{-1}\) exists

• A unique solution to \(A x=b\) (\(x^*=A^{-1}b\)) exists

Matrix Inversion

Adjoint of a nonsingular \(n \times n\) matrix: \[adj A = C^T = \begin{bmatrix} |C_{11}| & |C_{21}| & \cdots & |C_{n1}| \\ |C_{12}| & |C_{22}| & \cdots & |C_{n2}| \\ \vdots &\vdots & \cdots & \vdots \\ |C_{1n}| & |C_{2n}| & \cdots & |C_{nn}| \end{bmatrix}\]

The inverse of \(A\) is defined as: \[A^{-1} = \frac{1}{|A|} \, adj A\]

Find the Inverse

Solving a System of Equations

By matrix inversion: premultiply \(Ax = b\) by \(A^{-1}\) to get \[x^* = A^{-1} b\]

By Cramer’s rule: form \(A_k\) by replacing the \(k\)th column of \(A\) with \(b\); then \[x^*_k = \frac{|A_k|}{|A|}\]

If \(A\) is singular, a unique solution does not exist.

A Simple Economic Model

National income \(Y\) and consumption \(C\), where investment and government spending together are 30: \[Y = C + 30 \qquad C = 20 + \tfrac{1}{2} Y\]

Rearranged, this is \[Y - C = 30\] \[-Y + 2C = 40\]

Write it as \(Ax = b\). Solve it two ways: by matrix inversion, and by Cramer’s rule.

For your review

• Lecture slides and worksheets

• Practice problems 1–7 with solutions, also collected as one PDF per module

• Module notes for Linear Algebra

• Videos on finding the inverse, under Lecture 6

• Sample exam and help sheet, on the Exams page

• Additional problems in the textbook

Good luck!