Vectors & Matrices in Two Dimensions – Comprehensive Notes

Administrative & Context Notes

• Summer & Fall classes open for enrollment; scheduling reminder.
• Class: Precalculus – Week 15 (Vectors & Matrices in 2 D) — Jun 29
• Instructor: Annie Meyers; time slots given in ET/PT.

Review of 2-D Vectors

• Vectors viewed as instructions for a robot: 1st component → horizontal displacement; 2nd component → vertical displacement.
• Components written column-style \begin{pmatrix}x\y\end{pmatrix} or plain text “(x; y)”.
• Robot metaphor reinforces geometric meaning of addition & scalar multiplication.

Dot Product

• Definition for \mathbf{x}=\begin{pmatrix}x1\x2\end{pmatrix},\;\mathbf{y}=\begin{pmatrix}y1\y2\end{pmatrix}:
  $\mathbf{x}\,\cdot\,\mathbf{y}=x<em>1y</em>1+x<em>2y</em>2.$
• Properties
  • Commutative: $\mathbf{x}\cdot\mathbf{y}=\mathbf{y}\cdot\mathbf{x}.$
  • Scalar (real-valued) output.
• Self-dot gives squared magnitude: $|\mathbf{x}|^{2}=\mathbf{x}\cdot\mathbf{x}.$

Magnitude

• $|\mathbf{x}|=\sqrt{x<em>1^{2}+x</em>2^{2}}.$ Examples: |\begin{pmatrix}1\2\end{pmatrix}|=\sqrt5,\;|\begin{pmatrix}3\1\end{pmatrix}|=\sqrt{10}. (Used later for triangle inequality.)

Geometric Constructions with Two Given Basis Vectors

• Two specific vectors (diagrammed) named $\mathbf{x},\mathbf{y}.$
• Students practiced identifying: $2\mathbf{y},\;\mathbf{x}+\mathbf{y},\;2\mathbf{x}-\mathbf{y},\;\mathbf{x}+2\mathbf{y}.$ Key points:
  • Head-to-tail rule for addition.
  • Parallel scaling recognisable on grid lines drawn parallel to $\mathbf{x},\mathbf{y}.$
• Trick question: no arrow in diagram equalled $\mathbf{x}-\mathbf{y}.$

Linear Combinations

• General form in 2-D: $a\mathbf{x}+b\mathbf{y}.$ Scalars $a,b\in\mathbb R.$
• Example 1: $\mathbf{c}=\mathbf{x}+\mathbf{y}\Rightarrow (1)\mathbf{x}+(1)\mathbf{y}.$
• Example 2 (non-integer coefficients).
  • Target \mathbf{v}=\begin{pmatrix}5\4\end{pmatrix}.
  • Algebraic system \begin{cases}a+3b=5\2a+b=4\end{cases} solved to $a=\tfrac75,\;b=\tfrac65.$
  • Therefore $\mathbf{v}=\tfrac75\mathbf{x}+\tfrac65\mathbf{y}.$
• Example 3 (impossible case).
  • Showed \mathbf{w}=\begin{pmatrix}5\2\end{pmatrix} cannot be written from the set ${\mathbf{x},\,\mathbf{y},\,2\mathbf{y}}$ because all three candidate vectors are parallel; algebraic system inconsistent.

Using Linear Combinations in Dot-Product Problems

• To find angle between $\mathbf{v}=\mathbf{x}+\mathbf{y}$ and $\mathbf{w}=3\mathbf{x}-2\mathbf{y}$ given $|\mathbf{x}|=\sqrt5,\,|\mathbf{y}|=\sqrt{10},\,\mathbf{x}\cdot\mathbf{y}=5.$
• Algebraic expansion (linearity):
  $\mathbf{v}\cdot\mathbf{w}=(\mathbf{x}+\mathbf{y})\cdot(3\mathbf{x}-2\mathbf{y})=3\mathbf{x}\cdot\mathbf{x}-2\mathbf{x}\cdot\mathbf{y}+3\mathbf{y}\cdot\mathbf{x}-2\mathbf{y}\cdot\mathbf{y}.$ Simplifies to $0.$
• Hence $\cos\theta=0\;\Rightarrow\;\theta=90^{\circ}.$

Magnitude of a Linear Combination

• Computed $|\mathbf{v}|=|\mathbf{x}+\mathbf{y}|.$
  • Direct triangle inequality only gives an upper bound.
  • Dot-product route: $|\mathbf{v}|^{2}=(\mathbf{x}+\mathbf{y})\cdot(\mathbf{x}+\mathbf{y})=|\mathbf{x}|^{2}+2\mathbf{x}\cdot\mathbf{y}+|\mathbf{y}|^{2}=5+2\cdot5+10=25.$
  • Thus $|\mathbf{v}|=5.$

Projections

• Definition: Projection of $\mathbf{v}$ onto $\mathbf{u}$ (notation $\operatorname{proj}_{\mathbf{u}}\mathbf{v}$) is the “shadow” of $\mathbf{v}$ on a line having direction $\mathbf{u}$, with light rays perpendicular to that line.
• Facts observed geometrically:
  • If angle between $\mathbf{v},\mathbf{u}$ is acute ⇒ projection is positive multiple of $\mathbf{u}.$
  • Obtuse ⇒ negative multiple.
  • Right angle ⇒ zero vector.
  • If $\mathbf{v}$ is already parallel to $\mathbf{u}$ ⇒ projection equals $\mathbf{v}$ itself.
• Numerical Example: project \begin{pmatrix}2\3\end{pmatrix} onto \mathbf{u}=\begin{pmatrix}-1\0\end{pmatrix}.
  • Direction line horizontal (slope 0).
  • Projection found visually: \begin{pmatrix}-2\0\end{pmatrix} (opposite orientation still valid).
• Algebraic formula deferred to later lecture; focus kept on geometric intuition.

Introduction to Matrices

• Matrix = rectangular array of numbers.
  • Pronunciation: “MAY-trix”; plural “MAY-tri-seez”.
• Notation examples: \begin{pmatrix}1&2\3&4\end{pmatrix}\;\equiv\;(1,2;3,4).
• Dimensions (m × n): m rows (height), n columns (width).
• Rows/Columns treated as vectors: $\text{row}<em>1(A),\text{row}</em>2(A);\;\text{col}<em>1(A),\text{col}</em>2(A).$

Matrix Addition & Scalar Multiplication (2 × 2 only for now)

• Component-wise:
  \begin{pmatrix}1&-1\3&6\end{pmatrix}+\begin{pmatrix}-6&5\3&2\end{pmatrix}=\begin{pmatrix}-5&4\6&8\end{pmatrix}.
• Scalar example: 3\begin{pmatrix}1&2\3&4\end{pmatrix}=\begin{pmatrix}3&6\9&12\end{pmatrix}.

Matrix-Vector Multiplication

• Definition: For A=\begin{pmatrix}a&b\c&d\end{pmatrix},\;\mathbf{v}=\begin{pmatrix}v1\v2\end{pmatrix}
  A\mathbf{v}=\begin{pmatrix}(a,b)\cdot(v1,v2)\(c,d)\cdot(v1,v2)\end{pmatrix}=\begin{pmatrix}av1+bv2\cv1+dv2\end{pmatrix}. (Row-dot-vector rule.)
• Practice:
  • A=\begin{pmatrix}1&2\4&-1\end{pmatrix},\;\mathbf{e}1=\begin{pmatrix}1\0\end{pmatrix}\Rightarrow A\mathbf{e}1=\begin{pmatrix}1\4\end{pmatrix}.
  • Same matrix with \begin{pmatrix}2\1\end{pmatrix}\Rightarrow\begin{pmatrix}4\7\end{pmatrix}.

Important Special Matrices

• Identity I=\begin{pmatrix}1&0\0&1\end{pmatrix}
  • Acts as identity function: $I\mathbf{v}=\mathbf{v}\;\forall\mathbf{v}.$
• Rotation by $90^{\circ}$ counter-clockwise: J=\begin{pmatrix}0&-1\1&0\end{pmatrix}.
  • Maps $\begin{pmatrix}v<em>1\v</em>2\end{pmatrix}\mapsto\begin{pmatrix}-v<em>2\v</em>1\end{pmatrix}.$
  • Proven via correspondence with multiplying complex number $a+bi$ by $i$.
• Reflection across the line $y=x$ (direction vector \begin{pmatrix}1\1\end{pmatrix}): R=\begin{pmatrix}0&1\1&0\end{pmatrix}.
  • Swaps coordinates: $R\begin{pmatrix}v<em>1\v</em>2\end{pmatrix}=\begin{pmatrix}v<em>2\v</em>1\end{pmatrix}.$ Not a rotation (angle depends on input).

Conceptual View: Matrices as Functions

• Any 2 × 2 matrix defines a linear transformation $T:\mathbb R^{2}\to\mathbb R^{2},\;T(\mathbf{v})=A\mathbf{v}.$
• Examples:
  • $I$ → identity function.
  • $J$ → rotation by $\tfrac{\pi}{2}.$
  • $R$ → reflection about line in direction \begin{pmatrix}1\1\end{pmatrix}.

Additional Numerical & Diagram Insights

• Slope connection: direction vector $(-1,0)$ ⇒ horizontal line (slope 0).
• Direction vectors for lines used in projection tasks: any non-zero scalar multiple qualifies.
• Emphasis that vectors have no fixed location ⇒ rotations/reflections independent of centre, lines only need direction.
• Triangle Inequality revisited: $|\mathbf{x}+\mathbf{y}|\le|\mathbf{x}|+|\mathbf{y}|.$
• Perpendicular vectors have zero projection, dot product 0.
• Parallelism & scalar multiples: $k\mathbf{u}$ gives same direction for any non-zero $k\in\mathbb R.$

Key Vocabulary

• Vector, scalar, dot product, magnitude, linear combination, span (implicitly introduced), projection, direction vector, matrix, row, column, identity matrix, transformation, rotation, reflection.

Ethical / Practical Remarks

• Copyright notice: material protected; printing for personal use only.
• Reminder about class schedule break for July 4.

Takeaways & Next Steps

• Comfort with writing/solving linear combinations is foundational for future study (basis, span, linear independence upcoming).
• Geometric intuition of projection critical for later formula derivation and applications (least squares, components, work).
• Matrix arithmetic (especially multiplication) lays groundwork for solving systems, transformations and later eigenvalues.
• Upcoming classes will delve deeper into projections’ algebraic formula, conditions for expressing vectors as combinations, and broader linear algebra concepts.