Skip to main content

Change of Basis and Its Applications

The concept of changing the basis in a vector space is fundamental in linear algebra and has profound implications in data science, particularly in dimensionality reduction, data transformation, and simplifying complex problems. This article explores the concept of change of basis, the mathematical process behind it, and its practical applications.

1. What is a Basis?​

1.1 Definition of a Basis​

A basis of a vector space is a set of linearly independent vectors that span the entire space. Any vector in the space can be uniquely expressed as a linear combination of the basis vectors.

Example:​

In R2\mathbb{R}^2, the standard basis is:

e1=(10),e2=(01)\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad \mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}

Any vector v\mathbf{v} in R2\mathbb{R}^2 can be written as:

v=x1e1+x2e2=(x1x2)\mathbf{v} = x_1\mathbf{e}_1 + x_2\mathbf{e}_2 = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}

1.2 Why Change the Basis?​

Changing the basis allows us to express vectors in a new coordinate system that might simplify calculations, reveal hidden structures, or align with the problem's geometry. For example, in data science, changing the basis can align data with principal components (as in PCA), simplifying analysis and reducing dimensionality.

2. Mathematical Process of Changing the Basis​

2.1 Transition Matrix​

To change the basis from one set of basis vectors {b1,b2,…,bn}\{\mathbf{b}_1, \mathbf{b}_2, \dots, \mathbf{b}_n\} to another set {c1,c2,…,cn}\{\mathbf{c}_1, \mathbf{c}_2, \dots, \mathbf{c}_n\}, we use a transition matrix P\mathbf{P}.

If P\mathbf{P} is the matrix whose columns are the vectors of the new basis expressed in terms of the old basis, then for any vector v\mathbf{v}:

vnew=P−1vold\mathbf{v}_{\text{new}} = \mathbf{P}^{-1} \mathbf{v}_{\text{old}}

2.2 Example of Changing Basis​

Consider changing the basis in R2\mathbb{R}^2 from the standard basis {e1,e2}\{\mathbf{e}_1, \mathbf{e}_2\} to a new basis {b1,b2}\{\mathbf{b}_1, \mathbf{b}_2\} where:

b1=(21),b2=(13)\mathbf{b}_1 = \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \quad \mathbf{b}_2 = \begin{pmatrix} 1 \\ 3 \end{pmatrix}

The transition matrix P\mathbf{P} is:

P=(2113)\mathbf{P} = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}

To express a vector v=(57)\mathbf{v} = \begin{pmatrix} 5 \\ 7 \end{pmatrix} in the new basis, we calculate:

vnew=P−1v\mathbf{v}_{\text{new}} = \mathbf{P}^{-1} \mathbf{v}

where P−1\mathbf{P}^{-1} is the inverse of the transition matrix P\mathbf{P}.

3. Applications in Data Science​

3.1 Principal Component Analysis (PCA)​

In PCA, the data is transformed to a new basis where the axes (principal components) represent directions of maximum variance. Changing the basis to the principal components simplifies the data structure, often revealing more meaningful patterns and reducing dimensionality.

Example:​

After performing PCA on a dataset, the principal components form the new basis. The data can be projected onto this new basis, reducing the number of dimensions while retaining most of the variance.

3.2 Data Transformation​

In many machine learning algorithms, data is transformed to a new basis to improve the algorithm's performance or interpretability. For example, in linear discriminant analysis (LDA), data is projected onto a new basis that maximizes class separability.

3.3 Simplifying Linear Systems​

Changing the basis can simplify solving linear systems, particularly when the new basis aligns with the system's inherent structure. For instance, in diagonalization, changing the basis to the eigenvectors of a matrix simplifies matrix operations, making them easier to compute and understand.

4. Practical Considerations​

4.1 Choosing a Good Basis​

Choosing an appropriate basis is crucial. The new basis should align with the problem's structure or simplify the calculations. For example, in PCA, the principal components are chosen because they capture the most variance in the data.

4.2 Computational Efficiency​

Changing the basis, particularly in high-dimensional spaces, can be computationally expensive. Efficient algorithms and numerical methods are necessary to compute the transition matrices and transform the data effectively.

4.3 Interpreting Results​

When changing the basis, it's important to interpret the results correctly in the new coordinate system. For instance, after performing PCA, the transformed data represents projections on the principal components, which may require rethinking the original data's interpretation.

5. Connection to Other Linear Algebra Concepts​

5.1 Eigenvalues and Eigenvectors​

In many cases, the best basis for a problem is formed by the eigenvectors of a matrix. Changing the basis to the eigenvector basis simplifies many linear transformations, particularly those involving symmetric or diagonalizable matrices.

5.2 Orthogonality and Orthonormal Bases​

Changing to an orthonormal basis, where the basis vectors are orthogonal and of unit length, can greatly simplify many computations. This is particularly important in optimization and data science applications.

Conclusion​

The concept of changing the basis in vector spaces is a powerful tool in linear algebra, with numerous applications in data science. Whether you are performing dimensionality reduction, transforming data for machine learning, or simplifying linear systems, understanding how and when to change the basis is essential for effective problem-solving and data analysis.