Change Of Basis Of A Matrix

Let's delve into the fascinating world of linear algebra and explore the concept of the change of basis of a matrix. This is a fundamental idea that allows us to represent the same linear transformation in different coordinate systems, offering valuable insights and simplifying calculations. We'll cover the theoretical underpinnings, practical techniques, and some applications of this concept.

Understanding the Foundation: Vector Spaces and Bases

Before diving into the change of basis, let's solidify our understanding of the underlying concepts.

A vector space is a set of objects (called vectors) that can be added together and multiplied by scalars (usually real numbers) while still remaining within the same set. These operations must satisfy certain axioms, ensuring that the vector space behaves in a predictable and consistent manner. Examples of vector spaces include:

• The set of all n-tuples of real numbers, denoted as R^n (e.g., R^2 represents the familiar 2D plane).
• The set of all m x n matrices with real entries.
• The set of all polynomials with real coefficients.
• The set of all continuous functions defined on a given interval.

A basis for a vector space is a set of linearly independent vectors that span the entire space.

• Linear independence means that no vector in the set can be written as a linear combination of the other vectors. In other words, each vector contributes uniquely to the space.
• Spanning means that every vector in the space can be written as a linear combination of the vectors in the basis.

A crucial property of a basis is that it provides a unique way to represent any vector in the space. This unique representation is essential for performing calculations and understanding the structure of the vector space.

For example, the standard basis for R^2 is the set {(1, 0), (0, 1)}. Any vector (x, y) in R^2 can be written as a linear combination of these basis vectors: (x, y) = x(1, 0) + y(0, 1). Similarly, the standard basis for R^3 is {(1, 0, 0), (0, 1, 0), (0, 0, 1)}.

Linear Transformations and Their Matrix Representations

A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. That is, if T is a linear transformation from vector space V to vector space W, then:

• T(u + v) = T(u) + T(v) for all vectors u and v in V.
• T(cu) = cT(u) for all vectors u in V and all scalars c.

Linear transformations are fundamental in linear algebra because they describe how vectors are transformed in a structured and predictable way.

A crucial connection exists between linear transformations and matrices. Given a linear transformation T: V -> W, and given bases for V and W, we can represent T as a matrix. This matrix, often denoted as [T]B,C (where B is a basis for V and C is a basis for W), encodes all the information needed to apply the linear transformation.

The process of finding the matrix representation is as follows:

1. Choose a basis B = {b1, b2, ..., bn} for the vector space V.
2. Apply the linear transformation T to each basis vector bi.
3. Express each resulting vector T(bi) as a linear combination of the basis vectors in C = {c1, c2, ..., cm} for the vector space W.
4. The coefficients of these linear combinations form the columns of the matrix [T]B,C.

Therefore, the i-th column of [T]B,C consists of the coordinates of T(bi) with respect to the basis C.

Example:

Let T: R^2 -> R^2 be the linear transformation defined by T(x, y) = (2x + y, x - y). Let B = {(1, 0), (0, 1)} be the standard basis for R^2, and let C = {(1, 0), (0, 1)} also be the standard basis for R^2.

1. T(1, 0) = (2, 1) = 2(1, 0) + 1(0, 1)
2. T(0, 1) = (1, -1) = 1(1, 0) - 1(0, 1)

Therefore, the matrix representation of T with respect to the standard basis is:

[T]B,C = [[2, 1], [1, -1]]

The Change-of-Basis Matrix: Bridging Different Perspectives

The change-of-basis matrix is the cornerstone of understanding how to represent the same linear transformation in different coordinate systems. It allows us to translate the coordinates of a vector from one basis to another.

Let V be a vector space, and let B = {b1, b2, ..., bn} and *B' = {b1', b2', ..., bn'} be two different bases for V. The change-of-basis matrix from B to B', denoted as PB->B', is a matrix that transforms the coordinates of a vector with respect to the basis B into the coordinates of the same vector with respect to the basis B'.

Constructing the Change-of-Basis Matrix:

The j-th column of PB->B' consists of the coordinates of the basis vector bj (from basis B) with respect to the basis B'. In other words, to find PB->B', we need to express each vector in the basis B as a linear combination of the vectors in the basis B'.

Example:

Let B = {(1, 0), (0, 1)} be the standard basis for R^2, and let B' = {(1, 1), (1, -1)} be another basis for R^2. Let's find the change-of-basis matrix PB->B'.

1. Express (1, 0) in terms of (1, 1) and (1, -1): (1, 0) = (1/2)(1, 1) + (1/2)(1, -1)
2. Express (0, 1) in terms of (1, 1) and (1, -1): (0, 1) = (1/2)(1, 1) - (1/2)(1, -1)

Therefore, the change-of-basis matrix from B to B' is:

PB->B' = [[1/2, 1/2], [1/2, -1/2]]

Using the Change-of-Basis Matrix:

If v is a vector in V, and [v]B represents the coordinates of v with respect to the basis B, and [v]B' represents the coordinates of v with respect to the basis B', then:

[v]B' = PB->B' [v]B

This equation states that to find the coordinates of v with respect to B', you simply multiply the change-of-basis matrix PB->B' by the coordinates of v with respect to B.

Finding the Inverse Change-of-Basis Matrix:

The change-of-basis matrix from B' to B, denoted as PB'->B, is the inverse of PB->B'. That is:

PB'->B = (PB->B')^-1

This makes intuitive sense: if PB->B' transforms coordinates from B to B', then its inverse should transform coordinates from B' back to B.

In our previous example, PB->B' = [[1/2, 1/2], [1/2, -1/2]]. The inverse of this matrix is:

PB'->B = [[1, 1], [1, -1]]

You can verify that PB->B' PB'->B = I, where I is the identity matrix.

Change of Basis for a Matrix Representation of a Linear Transformation

Now, let's connect the change-of-basis matrix to the matrix representation of a linear transformation. Suppose we have a linear transformation T: V -> V (note that the domain and codomain are the same vector space). Let B and B' be two different bases for V. We want to find the relationship between the matrix representation of T with respect to B ([T]B) and the matrix representation of T with respect to B' ([T]B').

The key relationship is:

[T]B' = (PB->B') [*T*]B PB->B'

Where:

• [T]B is the matrix representation of T with respect to the basis B.
• [T]B' is the matrix representation of T with respect to the basis B'.
• PB->B' is the change-of-basis matrix from B to B'.
• (PB->B')^-1 is the change-of-basis matrix from B' to B.

This formula tells us that to find the matrix representation of T with respect to the new basis B', we need to:

1. Change the coordinates of the input vector from B' to B (using (PB->B')^-1).
2. Apply the linear transformation using the matrix representation with respect to B (using [*T*]B).
3. Change the coordinates of the output vector from B back to B' (using PB->B').

In simpler terms, we're sandwiching the original matrix representation between the change-of-basis matrix and its inverse. This process is called a similarity transformation. Two matrices A and B are said to be similar if there exists an invertible matrix P such that B = P^-1 A P.

Why is this useful?

Changing the basis can significantly simplify the matrix representation of a linear transformation. In some cases, we can find a basis B' such that [T]B' is a diagonal matrix. A diagonal matrix is much easier to work with than a general matrix, as its eigenvalues are simply the diagonal entries. This simplifies calculations involving eigenvalues, eigenvectors, and the long-term behavior of the linear transformation. The process of finding such a basis is called diagonalization.

Example: Change of Basis for a Linear Transformation

Let T: R^2 -> R^2 be the linear transformation defined by T(x, y) = (5x - y, 2x + 2y). Let B = {(1, 0), (0, 1)} be the standard basis for R^2, and let B' = {(1, 1), (1, 2)} be another basis for R^2.

1. Find the matrix representation of T with respect to B:

   • T(1, 0) = (5, 2) = 5(1, 0) + 2(0, 1)
   • T(0, 1) = (-1, 2) = -1(1, 0) + 2(0, 1)

   Therefore, [T]B = [[5, -1], [2, 2]]

2. Find the change-of-basis matrix from B to B':

   • (1, 0) = -2(1, 1) + 1(1, 2)
   • (0, 1) = 1(1, 1) - 1(1, 2)

   Therefore, PB->B' = [[-2, 1], [1, -1]]

3. Find the inverse of the change-of-basis matrix, which is PB'->B:

   PB'->B = (PB->B')^-1 = [[1, 1], [1, 2]]

4. Calculate the matrix representation of T with respect to B':

   [T]B' = (PB->B')^-1 [*T*]B PB->B' = [[1, 1], [1, 2]] [[5, -1], [2, 2]] [[-2, 1], [1, -1]] = [[1, 1], [1, 2]] [[-11, 6], [-2, 0]] = [[-13, 6], [-15, 6]]

Therefore, the matrix representation of T with respect to the basis B' is [T]B' = [[-13, 6], [-15, 6]].

Applications of Change of Basis

The change of basis is not just a theoretical concept; it has numerous practical applications in various fields:

• Computer Graphics: In computer graphics, objects are often represented using different coordinate systems. Changing the basis allows us to rotate, scale, and translate objects efficiently. For example, transforming an object to a coordinate system aligned with the camera simplifies the projection process.
• Data Analysis: In data analysis, principal component analysis (PCA) uses the change of basis to find a new set of axes (principal components) that capture the most variance in the data. This helps reduce the dimensionality of the data while preserving important information.
• Signal Processing: In signal processing, the Fourier transform can be viewed as a change of basis. It transforms a signal from the time domain to the frequency domain, allowing us to analyze the different frequency components of the signal.
• Solving Differential Equations: In some cases, a change of basis can simplify the process of solving differential equations. By transforming the equation to a new coordinate system, the equation may become easier to solve.
• Quantum Mechanics: In quantum mechanics, the state of a quantum system is represented by a vector in a Hilbert space. Changing the basis allows us to represent the same quantum state in different ways, which can be useful for calculations and interpretations. Diagonalizing the Hamiltonian (the operator representing the energy of the system) corresponds to finding the energy eigenstates of the system.

Common Questions and Considerations

• Is the change of basis unique?

  No, the change of basis is not unique. There are infinitely many different bases for a vector space, and therefore infinitely many different change-of-basis matrices between any two given bases.

• What happens if the bases are not linearly independent?

  If the bases are not linearly independent, then the change-of-basis matrix will not be invertible. This means that it's not possible to uniquely transform coordinates between the two sets of vectors. Therefore, it's crucial that both B and B' are indeed bases for V.

• How do I find a "good" basis?

  The choice of a "good" basis depends on the specific problem. In some cases, the standard basis is the most convenient. In other cases, a different basis may simplify calculations or provide more insight into the problem. Diagonalization aims to find a basis that simplifies the matrix representation of a linear transformation as much as possible.

Conclusion

The change of basis is a powerful concept in linear algebra that allows us to represent vectors and linear transformations in different coordinate systems. It provides a bridge between different perspectives and can significantly simplify calculations. By understanding the underlying theory and practical techniques, you can unlock new insights and solve complex problems in various fields. Mastering this concept is essential for anyone working with linear algebra, whether in mathematics, computer science, engineering, or other related disciplines. The ability to flexibly change viewpoints provides a significant advantage in understanding and manipulating linear systems.