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Exam details
\[\mathbf{x} = \begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{bmatrix} \in \mathbb{R}^n\]
Symbols: \(x_i\) = i-th component; \(\mathbb{R}^n\) = n-dimensional real space.
Intuition: A vector is an ordered list of n numbers. In ML it represents a data point, feature embedding, or parameter set as a point in n-dimensional space.
\[\mathbf{a} + \mathbf{b} = \begin{bmatrix}a_1+b_1 \\ a_2+b_2 \\ \vdots \\ a_n+b_n\end{bmatrix}\]
Symbols: \(a_i, b_i\) = corresponding components of vectors a and b.
Intuition: Add element-wise. Geometrically, place the tail of b at the head of a; the sum points to the new head.
\[c\mathbf{x} = \begin{bmatrix}cx_1 \\ cx_2 \\ \vdots \\ cx_n\end{bmatrix}\]
Symbols: \(c \in \mathbb{R}\) = scalar; \(x_i\) = i-th component.
Intuition: Scales every component by c. Stretches (|c|>1), shrinks (|c|<1), or flips direction (c<0) the vector.
\[\|\mathbf{x}\|_2 = \sqrt{\sum_{i=1}^n x_i^2}\]
Symbols: \(\|\cdot\|_2\) = L2 norm; \(x_i\) = i-th component; \(n\) = dimension.
Intuition: The straight-line (Euclidean) distance from the origin to the tip of the vector. Most common norm in ML for measuring distances.
\[\|\mathbf{x}\|_1 = \sum_{i=1}^n |x_i|\]
Symbols: \(|x_i|\) = absolute value of i-th component.
Intuition: Sum of absolute values. Also called the 'Manhattan' or 'taxicab' distance. Encourages sparsity in optimization (L1 regularization).