Change of Basis in Vector Space

by Kurious Fox

Applications of Change of Basis in Vector Space in linear algebra are fundamental, allowing for a transformative approach to understanding and manipulating vectors. This technique is particularly useful when dealing with complex problems, as it enables the representation of vectors in different coordinate systems, which can simplify calculations and interpretations. By changing the basis of a vector space, mathematicians and scientists can express linear combinations of vectors more intuitively, optimize solutions to linear equations, and facilitate easier computations in higher-dimensional spaces. Moreover, this concept plays a crucial role in areas such as computer graphics, where transformations and rotations of objects are essential, as well as in the field of machine learning, particularly in feature transformation and dimensionality reduction methods like Principal Component Analysis (PCA).

Change of Basis:

To change the basis, you typically need to express the existing vectors in terms of a new set of basis vectors, which may involve calculating a transition or change of basis matrix. This matrix is derived from the coordinates of the new basis vectors compared to the old ones and is critical for transforming any vector from one basis to another. After that, we multiply the transition matrix with the original vector to obtain the vector representations in the new basis. Specifically, to transform a vector v from basis B to another basis C :

  1. Express v in terms of B : v_B .
  2. Use the transition matrix P_{B \to C} :
    v_C = P_{B \to C} v_B

If B = {b_1, b_2, \dots} and C = {c_1, c_2, \dots} , the columns of P_{B \to C} are the coordinates of c_i in basis B .

Let’s go through an example of changing the basis in a vector space. We’ll consider the vector space \mathbb{R}^2 and change from one basis to another.

Given:

  1. Original Basis B: \mathbf{b}_1 = \begin{pmatrix} 1 \\0 \end{pmatrix} , \mathbf{b}_2 = \begin{pmatrix} 0 \\1 \end{pmatrix}
  2. New Basis C: \mathbf{c}_1 = \begin{pmatrix} 1 \\1 \end{pmatrix} , \mathbf{c}_2 = \begin{pmatrix} 1 \\-1 \end{pmatrix}
  3. Vector in Original Basis: \mathbf{v} = \begin{pmatrix} 3 \\2 \end{pmatrix}

Step-by-Step Process to Find the Transition Matrix and Change the Basis:

Form the Matrix of the New Basis C :
Construct matrix C using the new basis vectors \mathbf{c}_1 and \mathbf{c}_2 :
C = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}

Form the Matrix of the Original Basis B :
Construct matrix B using the original basis vectors \mathbf{b}_1 and \mathbf{b}_2 :
B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

Find the Inverse of Matrix C :
Calculate the inverse of C :
C^{-1} = \frac{1}{\det(C)} \begin{pmatrix} -1 & -1 \\ -1 & 1 \end{pmatrix}
\det(C) = 1 \cdot (-1) - 1 \cdot 1 = -2 \Rightarrow C^{-1} = \frac{1}{-2} \begin{pmatrix} -1 & -1 \\ -1 & 1 \end{pmatrix} = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & -0.5 \end{pmatrix}

Construct the Transition Matrix P :
The transition matrix from basis B to basis C is given by:
P = C^{-1}
So,
P = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & -0.5 \end{pmatrix}

Convert the Vector \mathbf{v} to the New Basis:
To express \mathbf{v} in the new basis C , multiply it by P :
\mathbf{v}_C = P \mathbf{v} = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & -0.5 \end{pmatrix} \begin{pmatrix} 3 \\ 2 \end{pmatrix} = \begin{pmatrix} 0.5 \cdot 3 + 0.5 \cdot 2 \\ 0.5 \cdot 3 - 0.5 \cdot 2 \end{pmatrix} = \begin{pmatrix} 2.5 \\ 0.5 \end{pmatrix}

Conclusion: The coordinates of the vector \mathbf{v} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} in the new basis C are \begin{pmatrix} 2.5 \\ 0.5 \end{pmatrix} using the transition matrix method.

Step-by-Step Process without using a transition matrix:

The following equivalent step-by-step process without using a transition matrix may help you understand a bit better the process with a transition matrix. This is due to the relation between matrix inversion and solving a linear system of equations.

Write the Vector in Terms of the New Basis C:

  • We need to express \mathbf{v} in terms of the new basis vectors \mathbf{c}_1 and \mathbf{c}_2 .

Set Up the Equations:

  • Let \mathbf{v} be a\mathbf{c}_1 + b\mathbf{c}_2 .
  • So, \begin{pmatrix} 3 \\2 \end{pmatrix} = a \begin{pmatrix} 1 \\1 \end{pmatrix} + b \begin{pmatrix} 1 \\-1 \end{pmatrix} .

Form the System of Linear Equations:

  • \begin{pmatrix} 3 \\2 \end{pmatrix} = a \begin{pmatrix} 1 \\1 \end{pmatrix} + b \begin{pmatrix} 1 \\-1 \end{pmatrix}
  • This gives us:
    3 = a + b
    2 = a - b

Solve the System:

  • Add the equations:
    3 + 2 = (a + b) + (a - b)
    5 = 2a
    a = 2.5
  • Substitute a back into one of the original equations:
    3 = 2.5 + b
    b = 3 - 2.5
    b = 0.5

Vector in New Basis:

  • So, the vector \mathbf{v} in terms of the new basis \mathbf{c}_1 and \mathbf{c}_2 is \mathbf{v} = 2.5\mathbf{c}_1 + 0.5\mathbf{c}_2 .

Conclusion:

  • The coordinates of the vector \mathbf{v} in the new basis C are \begin{pmatrix} 2.5 \\0.5 \end{pmatrix} .

This example illustrates how to change the basis of a vector in \mathbb{R}^2 . By solving the system of linear equations, we can find the new coordinates of the vector on the desired basis.

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