Mathematics IB Algebra Course Notes

Mathematics IB Algebra Course Notes

Course Introduction

• Written by: Prof. Michael Murray

• Version: 2025

Chapter 1: Bases and Dimension

1.1 Review of Mathematics IA

• Key definitions and concepts of Mathematics IA.

• Subspace of $extbf{R}^n$: A subset $V$ of $extbf{R}^n$ such that:

  • The zero vector $0$ is in $V$.

  • If $u, v ext{ in } V$, then $u + v ext{ in } V$.

  • If $u ext{ in } V$ and $c ext{ in } extbf{R}$, then $cu ext{ in } V$.

• Linear Combination: A vector $w$ is a linear combination of $v<em>1, ext{…}, v</em>k$ if $w = c<em>1v</em>1 + c<em>2v</em>2 + … + c<em>kv</em>k$ for some $c<em>1, … , c</em>k ext{ in } extbf{R}$.

• Linearly Independent Vectors: Vectors $v<em>1, …, v</em>k$ are linearly independent if the only solution to $0 = c<em>1v</em>1 + c<em>2v</em>2 + ext{…} + c<em>kv</em>k$ is $c<em>1 = c</em>2 = … = c_k = 0$.

• Span: The set of all linear combinations of $v<em>1, … , v</em>k$ is called the span of $v<em>1, … , v</em>k$ denoted $ext{span} ext{{ t extbf{v}}<em>1, …, extbf{v}</em>k}$.

• Spanning Set: We say $v<em>1, …, v</em>k$ span $V$ if $ext{span} ext{{ t extbf{v}}<em>1, … , extbf{v}</em>k} = V$.

• Basis of V: A linearly independent set $ext{{ t extbf{v}}<em>1, … , extbf{v}</em>k}$ that spans $V$ is called a basis for $V$.

Example 1.1

• Linear Dependence: Any list of vectors that includes $0$ is linearly dependent.

• Zero Subspace: The set $ext{{ t extbf{ ext{0}}}}$ is called the zero subspace; it does not have a basis.

1.2 Transpose

• Definition: If A is an $m imes n$ matrix then

ext{Transposed } A^t = egin{pmatrix} ext{rows become columns and columns become rows} ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{…} ext{ } ext{…} ext{ } ext{…} ext{ } ext{ } ext{ } ext{} ext{…} ext{ } ext{…} ext{ } ext{…} ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{…} ext{ } ext{…} ext{ } ext{…} ext{ } ext{ } ext{ } ext{ }} ext{ } ext{} ext{ } ext{ }egin{pmatrix} a{11} & a{21} &… \ a{12} & a{22} &… \ ext{…} & ext{…} & … ext{ } ext{ }} ext{ } ext{} ext{ } ext{ }egin{pmatrix} a{1n} & a{2n} &… ext{ }\ ext{…} & ext{…} & … ext{ } ext{}}\ ext{ } ext{…} ext{ } ext{ } ext{} ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{} ext{ }\ ext{ } ext{…} ext{ } ext{ } ext{} ext{ } ext{ } ext{ } ext{ } ext{ } ext{} ext{ }\ … … ext{…} ext{…} ext{…} ext{ }} ext{… } ext{… } ext{…} ext{ } ext{ } ext{} ext{ } ext{…} ext{…} ext{…}

Theorems

1. (A^t)^t = A.

2. If $A$ is a $k imes n$ matrix and $B$ is an $n imes m$ then (AB)^t = B^t A^t.

3. If $ext{det}(A) = ext{det}(A^t)$ for square matrices.

4. For an invertible matrix, (A^t)^{-1} = (A^{-1})^t.

Chapter 2: Row Space, Null Space and Column Space

2.1 Row Space

• The row space of a matrix A is the span of its rows.

• Example 1: If A = egin{pmatrix} 1 & 2 \ 3 & 4 \ 5 & 6 ext{ }, ext{…} ext{ } ext{ } ext{…} ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{} ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } (R^n)

Theorems & Definitions

• Null Space: $ext{Nul}(A) = ext{space of solutions to } Ax = 0.$

• Column Space: $ext{Col}(A) = ext{span of columns of } A.$

• The Rank Theorem: $ext{dim}( ext{Nul}(A)) + ext{dim}( ext{Row}(A)) = n$

• Kernel of A: $K(A) = ext{{Null}}(A)$

• Range of A: $R(A) = ext{{Col}}(A)$

Chapter 3: Linear Transformations

Definitions

• Linear Transformation: A function $F: ext{R}^n ightarrow ext{R}^m$ is linear if:

  1. $F(v + u) = F(v) + F(u), orall v, u ext{ in } ext{R}^n$ (Addition)

  2. $F(ku) = kF(u)$ for $k ext{ in } extbf{R}$.

Properties

• The kernel of a linear transformation $F$ is $ext{ker}(F) = ext{Nul}(A)$

• The range of $F$ is $R(F) = ext{Col}(A)$

Theorems

• The kernel and range of a linear transformation are both subspaces.

Chapter 4: Orthogonality, Gram-Schmidt

Definitions and Key Concepts

• Inner Product: u ullet v = u1v1 + ext{…} + unvn

• Length of a vector: ||u|| = ext{sqrt}(u ullet u)

• Orthogonality Definition: If u ullet v = 0, then $u$ and $v$ are orthogonal.

Gram-Schmidt Process

• Method to convert a basis into an orthonormal basis.

Applications

• Orthogonal projection onto a subspace, linear regression, etc.