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Change Of Coordinates Matrix

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The Change of Coordinates Matrix: A Bridge Between Coordinate Systems



Linear algebra provides powerful tools for navigating different perspectives on the same vector space. One such tool is the change of coordinates matrix, which allows us to seamlessly transition between representations of vectors expressed in different bases. This article will delve into the mechanics and significance of this matrix, illustrating its utility with clear examples and explanations.

1. Understanding Coordinate Systems and Bases



Before exploring the change of coordinates matrix, let's establish a firm understanding of coordinate systems and bases. A coordinate system provides a way to represent vectors uniquely using a set of scalar values. These values are determined by expressing the vector as a linear combination of basis vectors. A basis for a vector space V is a linearly independent set of vectors that spans V – meaning any vector in V can be written as a unique linear combination of these basis vectors.

For example, in the familiar 2D Cartesian plane, the standard basis is { i = (1, 0), j = (0, 1) }. The vector v = (3, 2) can be expressed as 3i + 2j. Here, 3 and 2 are the coordinates of v with respect to the standard basis. However, we can choose different bases to represent the same vector.

2. Constructing the Change of Coordinates Matrix



The core idea behind the change of coordinates matrix lies in expressing the vectors of one basis in terms of another. Suppose we have two bases for a vector space: B = {b₁, b₂, ..., bₙ} and B' = {b'₁, b'₂, ..., b'ₙ}. To find the change of coordinates matrix from B to B', denoted as `P_(B'←B)`, we need to express each vector in B as a linear combination of vectors in B'.

Let's represent the vectors of B' as column vectors. Then, if we express each vector of B as a linear combination of B' vectors, we get:

b₁ = c₁₁b'₁ + c₂₁b'₂ + ... + cₙ₁b'ₙ
b₂ = c₁₂b'₁ + c₂₂b'₂ + ... + cₙ₂b'ₙ
...
bₙ = c₁ₙb'₁ + c₂ₙb'₂ + ... + cₙₙb'ₙ

The coefficients `cᵢⱼ` form the entries of the change of coordinates matrix `P_(B'←B)`:

```
P_(B'←B) = [ c₁₁ c₁₂ ... c₁ₙ ]
[ c₂₁ c₂₂ ... c₂ₙ ]
[ ... ... ... ]
[ cₙ₁ cₙ₂ ... cₙₙ ]
```

This matrix transforms the coordinate vector of a vector in basis B to its coordinate vector in basis B'.

3. Applying the Change of Coordinates Matrix



Let's consider a concrete example. Suppose we have two bases in R²:

B = { (1, 0), (0, 1) } (standard basis)
B' = { (1, 1), (1, -1) }

To find `P_(B'←B)`, we express each vector in B in terms of B':

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

Therefore:

`P_(B'←B) = [ 1/2 1/2 ]
[ 1/2 -1/2 ]`

Now, let's say we have a vector v = (3, 2) in the standard basis B. Its coordinate vector is [3, 2]ᵀ. To find its coordinates in B', we multiply the change of coordinates matrix by the coordinate vector in B:

[3, 2]ᵀ x `P_(B'←B)` = [5/2, 1/2]ᵀ

Thus, the coordinates of v in B' are (5/2, 1/2).

4. Inverse Matrix and Change of Coordinates in the Opposite Direction



The change of coordinates matrix `P_(B←B')` (from B' to B) is simply the inverse of `P_(B'←B)`. This is intuitive since converting from B to B' and then back to B should return the original coordinates. We can verify this using our example:

`P_(B'←B)`⁻¹ = [ 1 1 ]
[ 1 -1 ]


5. Conclusion



The change of coordinates matrix is a fundamental concept in linear algebra that facilitates transitions between different representations of vectors within the same vector space. Its construction and application are straightforward, yet powerful, allowing for flexibility in choosing a basis that best suits the problem at hand. Understanding this concept is crucial for various applications, including computer graphics, physics, and engineering.


FAQs:



1. What happens if the bases are not linearly independent? You cannot form a basis with linearly dependent vectors. The change of coordinates matrix will not be defined.

2. Can the change of coordinates matrix be applied to matrices? No, directly. It applies to coordinate vectors representing vectors. However, the same principle can be extended to transform matrices representing linear transformations.

3. What if I have more than two bases? You can chain change-of-coordinates matrices. To go from B₁ to B₃, you'd first go from B₁ to B₂, then from B₂ to B₃.

4. Is the change of coordinates matrix always square? Yes, because the dimensions of the two bases must be the same for them both to be bases of the same vector space.

5. What are some real-world applications? Change of coordinates matrices are essential in computer graphics (transformations), quantum mechanics (changing representations of quantum states), and signal processing (representing signals in different frequency domains).

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