Vectors & Matrices in Two Dimensions – Comprehensive Notes

Administrative & Context Notes

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.

Definition for \mathbf{x}=\begin{pmatrix}x1\x2\end{pmatrix},\;\mathbf{y}=\begin{pmatrix}y1\y2\end{pmatrix}:
\mathbf{x}\,\cdot\,\mathbf{y}=x1y1+x2y2.
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}.

|\mathbf{x}|=\sqrt{x1^{2}+x2^{2}}. Examples: |\begin{pmatrix}1\2\end{pmatrix}|=\sqrt5,\;|\begin{pmatrix}3\1\end{pmatrix}|=\sqrt{10}. (Used later for triangle inequality.)

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}.

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.

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}.

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.

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.

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}1(A),\text{row}2(A);\;\text{col}1(A),\text{col}2(A).

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}.

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}.

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}v1\v2\end{pmatrix}\mapsto\begin{pmatrix}-v2\v1\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}v1\v2\end{pmatrix}=\begin{pmatrix}v2\v1\end{pmatrix}. Not a rotation (angle depends on input).

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}.

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.

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

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.