Chapter Summary

2.7. Chapter Summary#

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2.7.1. Key Take-aways:#

Introduction to Vectors

  • A vector is an ordered collection of numbers with accompanying mathematical operations.

    • Purpose: Vectors represent quantities with magnitude and direction, and store collections of numerical data in science and data analysis.

    • Notation: Written as bold lowercase letters (e.g., \(\mathbf{u}\))

    • Representation: Typically shown as a column of numbers enclosed in square brackets, like \begin{equation*} \mathbf{u} = \begin{bmatrix} 0.75\ -1\ 1.75\ 2.5 \end{bmatrix} \end{equation*}

    • or horizontally with transpose: \(\begin{bmatrix} 0.75 ~~-1 ~~1.75 ~~2.5\end{bmatrix}^{\text{T}}\)

    • Components: Individual values accessed via indices (e.g., \(u_i\) for the \(i\)th component)

  • A scalar is a single numeric value; scalar variables are represented by italicized letters, like \(c\).

  • Characterization of vectors

    • Order: The number of axes needed to represent a mathematical object. Vectors are order 1, and scalars are order 0.

    • Size/dimension: Number of components in a vector. We say a vector with \(n\) components is an \(n\)-dimensional vector or just \(n\)-vector

  • Implementation

    • Using NumPy, the most direct way to create a vector is to pass a list of values to the np.array() class constructor, like u = np.array([1, 4, 9]). Such a vector is neither explicitly a row-vector nor a column vector, but acts like a row vector in most NumPy operations.

    • Using NumPy, row vectors can be created like u = np.array([[1, 4, 9]]) – note the double square brackets. Column vectors are most easily created like u = np.array([[1, 4, 9]]).T.

    • To create vectors in PyTorch, modify any of the above NumPy calls by replacing np.array by torch.tensor. For instance, u2 = torch.tensor([[1, 4, 9]])

Visualizing Vectors

  • Vectors are commonly displayed using quiver plots that show the vectors as arrows that use the components of the vector as displacements; if no starting point is specified for a vector, the origin is used.

    • In 2-vectors, components represent \(x\) and \(y\) displacements: \(\mathbf{a} = \begin{bmatrix} a_x & a_y \end{bmatrix}.T\)

    • Each vector has a tail (starting point) and a head (endpoint), with the arrow pointing to the head.

Applications of Vectors

  • Vectors can be used for many purposes, including:

    • Geometrical features: Vectors represent spatial location or displacement. Example: A robot’s movements shown as arrows, where consecutive vectors indicate path segments.

    • Multi-dimensional numerical data: Vectors store collections of related numerical values. Example: Health survey data with height and weight.

    • Time-series data: Vectors represent observations collected over time at regular intervals. Example: Annual temperature measurements or monthly average house prices in an area.

    • Distributional data: Vectors represent allocations across categories or proportional distributions. Example: stock market portfolio allocations.

  • Other visualization techniques

    • Scatter plots are often used for large 2D data sets (less clutter than quiver plots)

    • Bar charts are often used for distributional data

Special Vectors

  • Zero vector: A vector containing all zeros, denoted as \(\mathbf{0}_n\) for a vector of size \(n\). In NumPy, created using np.zeros(n).

  • Ones vector: A vector containing all ones, denoted as \(\mathbf{1}_n\) for a vector of size \(n\). In NumPy, created using np.ones(n).

  • Standard unit vector: A vector with all components equal to zero except one element that equals one. Denoted as \(\mathbf{e}_i\) where \(i\) is the position of the 1.

Vector Operations

  • Vector addition: Component-wise addition: \((\mathbf{a}+\mathbf{b})_i = a_i + b_i\).

  • Scalar-vector multiplication: Each component is multiplied by that scalar: \((\alpha \mathbf{u})_i = \alpha u_i\).

  • Vector subtraction: Component-wise subtraction: \((\mathbf{a}-\mathbf{b})_i = a_i - b_i\).

  • Hadamard (component-wise) product: Denoted as \(\mathbf{u} \odot \mathbf{v}\), multiplies corresponding elements. In NumPy, uses the * operator.

  • Dot product: Multiplication between two vectors that produces a scalar: \(\mathbf{u} \cdot \mathbf{v} = \sum_{i=0}^{n-1} u_i v_i\). In NumPy, calculated using the @ operator.

  • Vector norm/length: Length of a vector defined as \(\|\mathbf{b}\| = \sqrt{\mathbf{b} \cdot \mathbf{b}} = \sqrt{\sum_{i=0}^{n-1} b_i^2}\). In NumPy, calculated using np.linalg.norm().

  • Unit vectors: Vectors with length 1, created by dividing a vector by its norm.

  • Distance between vectors: The norm of their difference: \(\operatorname{d}(\mathbf{a},\mathbf{b}) = \|\mathbf{a}-\mathbf{b}\|\).

Self-Assessment:

The following question can be used to check your understanding of the material covered in Chapter 2 of Linear Algebra for Data Science with Python. Reload this page to get a different selection of practice questions.

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