Vector Algebra

Vectors

A vector \(\mathbf{v} \in \mathbb{R}^n\) is an ordered list of \(n\) real numbers:

\[\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}\]

Basic Operations

Addition: \(\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_1 + v_1 \\ \vdots \\ u_n + v_n \end{pmatrix}\)

Scalar multiplication: \(c\mathbf{v} = \begin{pmatrix} cv_1 \\ \vdots \\ cv_n \end{pmatrix}\)

Dot Product

\[\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^{n} u_i v_i = \|\mathbf{u}\|\|\mathbf{v}\|\cos\theta\]

• \(\mathbf{u} \cdot \mathbf{v} = 0 \Rightarrow\) vectors are orthogonal
• Used to compute projections, angles, and similarity

Norms

The Euclidean norm (length) of \(\mathbf{v}\):

\[\|\mathbf{v}\| = \sqrt{\mathbf{v} \cdot \mathbf{v}} = \sqrt{\sum_{i=1}^n v_i^2}\]

A unit vector: \(\hat{\mathbf{v}} = \dfrac{\mathbf{v}}{\|\mathbf{v}\|}\)

Linear Combination and Span

A linear combination of vectors \(\mathbf{v}_1, \ldots, \mathbf{v}_k\):

\[c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k\]

The span is the set of all possible linear combinations. Vectors are linearly independent if no vector in the set can be written as a linear combination of the others.

---

Next: Linear Transformations