[Chapter 1 Summary] of Linear Algebra

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Chapter 1: Linear Equations

1. Systems of Linear Equations

• Equivalence: System $\iff$ Solution Sets $\to$ Row equivalent.
• Consistency:
  • Consistent $\leftarrow$ One solution / Infinite solutions.
  • Inconsistent $\leftarrow$ No solution.
• Matrix Representations:
  • Coefficient Matrix
  • Augmented Matrix

1.1 Row Reduction & Existence

• Elementary Row Operations: (1) Replacement, (2) Interchange, (3) Scaling.
• Forms: Echelon Form $\to$ Reduced Echelon Form (Uniqueness).
• Existence and Uniqueness:
  • No solution if the system contains a row like $[0 \dots 0 \ b]$ where $b \neq 0$.
  • Pivot determines basic variables and free variables.
  • Free variables imply infinite solutions (uniqueness fails).

2. Vector Equations

• Notation: Column vector $(a, b)$ or $\begin{bmatrix} a \\ b \end{bmatrix}$.
• Linear Combination: $y = c_1v_1 + c_2v_2 + \dots + c_nv_n \Rightarrow [a_1, a_2 \dots a_n \ b]$, where $c_i$ are the weights.
• Span: $\forall c \Rightarrow \text{Span}\{v_1 \dots v_p\}$.

3. Matrix Equation

• Form: $Ax = b$, where $A$ is an $m \times n$ matrix and $b \in \mathbb{R}^m$.
• Existence of Solutions: $b$ is a linear combination of all columns of $A$.

  $$Ax = [a_1, a_2 \dots a_n] \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix} = x_1a_1 + x_2a_2 + \dots + x_na_n$$

• Row-vector Rule for Computing $Ax$: Products of Row $i$ in $A$ and vector $x$.

  $$\begin{bmatrix} 2 & 3 & 4 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = [2x_1 + 3x_2 + 4x_3]$$

• Properties:

  $$A(u+v) = Au + Av$$

  $$A(cu) = c(Au)$$

4. Solution Sets

4.1 Homogeneous Systems ($Ax = 0$)

• At least a trivial solution exists.
• Nontrivial solution $\to$ free variable exists.

4.2 Nonhomogeneous Systems ($Ax = b$)

• General solution form: $\vec{x} = \vec{p} + t\vec{v}$, where $\vec{p}$ is a particular solution.
• Parametric Vector Form: $\vec{x} = s\vec{u} + t\vec{v}$.

5. Linear Independence

• Definition: Only trivial solution exists. If not, it has a linear dependence relation $c_1v_1 + c_2v_2 + \dots + c_nv_n = 0$ (where $c$ not all zero).
• Matrix Columns: Linear Independence of Matrix Columns $\iff$ At least 1 in $S$ is a linear combination of others.
• Rules of Thumb:
  • Vectors > Entries in each vector $\to$ linearly dependent.
  • Zero vector $\to$ linearly dependent.

Linear Transformation

1. Introduction to Linear Transformations

• Mapping Notation: $T : \mathbb{R}^n \to \mathbb{R}^m$
  • Domain $\to$ Codomain
  • $T(x) \to$ Range (Image of $x$)
• Matrix Transformation: $x \mapsto Ax$.
• Conditions for Linear Transformation (iff):

  $$T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$$

  $$cT(\vec{u}) = T(c\vec{u})$$

  • Implication: $T(0) = 0$
• Simplified Form:

  $$T(c\vec{u} + d\vec{v}) = cT(\vec{u}) + dT(\vec{v})$$

2. The Matrix of a Linear Transformation

• $T: \mathbb{R}^n \to \mathbb{R}^m \implies$ Unique $A : Ax = T(x)$.
• Standard Matrix: $A = [T(e_1), T(e_2) \dots T(e_n)]$, where $e_j$ is the $j$-th column of the Identity Matrix.

3. Mapping Properties (Onto & One-to-one)

Mapping $T: \mathbb{R}^n \to \mathbb{R}^m$ can be classified as follows:

Pivot Summary

A pivot in every:

$$\begin{cases} \text{row} \to \text{at least 1 solution} \to \text{Onto} \\ \text{column} \to \text{at most 1 solution} \to \text{One-to-one} \end{cases}$$