Exploring Change of Basis Matrices and Intuitive Examples in Linear Algebra

In summary: Similarly, (0,1) = 0.8 (1,-1) + 1/4(1,1) so the first row of the matrix is (1/4 1/4). The second row is (1.1), because the first basis vector (0,1) is multiplied by the (1.1)th row of the matrix.
  • #1
aaaa202
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Given a basis A = {a1,a2...an} we can always translate coordinates originally expressed with this basis to another basis A' = {a1',a2'...an'}. To do this we simply do some matrix-multiplication and it turns out that the change of basis matrix equals a square matrix whose rows are the coordinates of the original basis vectors written in terms of the new basis-vectors. I'm finding this a little hard to understand intuitively - can someone give me an example from maybe R^2 that shows why this is in an intuitive manner.
 
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  • #2
aaaa202 said:
that the change of basis matrix equals a square matrix whose rows are the coordinates of the original basis vectors written in terms of the new basis-vectors.

That depends on how we define a "change of basis matrix". For example if I have the row vector [itex] V_a [/itex] in basis [itex] A [/itex] and [itex]S [/itex] is "the change of basis matrix from basis [itex] A [/itex] to basis [itex] B [/itex] then how do I find that vector expessed in basis [itex] B [/itex]?

Do I compute [itex] (V_a)( S) [/itex] or do I compute [itex] (S)(V_a)^T [/itex] ? It depends on whether the course materials like to treat vectors as row vectors or as column vectors.

You should be able to construct a 2x2 example.

Let
[itex] a_1 = S_{11} b_1 + S_{12} b_2 [/itex]
[itex] a_2 = S_{21} b_1 + S_{22} b_2 [/itex]

What is the vector [itex] 4 a_1 + 3 a_2 [/itex] equal to in the [itex] B [/itex] basis?
 
  • #3
You can factor out transformations in terms of rotations, translations, and scaling operations.

By composing maps, you can see geometrically what the translation is doing by treating each map separately in terms of its geometric effect on a vector.
 
  • #4
aaaa202 said:
Given a basis A = {a1,a2...an} we can always translate coordinates originally expressed with this basis to another basis A' = {a1',a2'...an'}. To do this we simply do some matrix-multiplication and it turns out that the change of basis matrix equals a square matrix whose rows are the coordinates of the original basis vectors written in terms of the new basis-vectors. I'm finding this a little hard to understand intuitively - can someone give me an example from maybe R^2 that shows why this is in an intuitive manner.

Each vector in the first basis can be written as a linear combination of the vectors in the second. The coefficients of this linear combination are a column in the change of basis matrix.
The set of columns, one for each basis vector in the first basis, make up the change of basis matrix.
Multiplying a basis vector by the matrix just writes the vector out in terms of the second basis.

E.g. take for the first basis of R^2 the vectors (1,0) and (0,1) and the second basis (1,-1) and (1.1).

Then (1,0) = 1.2 (1,-1) + 1/2(1,1) so the first column of the matrix is (1/2 1/2).

The second column is (-1/2 1/2)
 
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  • #5


The concept of change of basis matrices in linear algebra can be a bit abstract, but let's use an example in R^2 to help make it more intuitive.

Imagine we have two different bases in R^2 - the standard basis, which consists of the vectors (1,0) and (0,1), and a new basis, which consists of the vectors (2,1) and (-1,2).

Now, let's say we have a point in R^2 with coordinates (3,4) in the standard basis. This means that the point is 3 units in the x-direction and 4 units in the y-direction. However, if we want to express this point in the new basis, we need to find the coordinates in terms of the (2,1) and (-1,2) vectors.

To do this, we can use a change of basis matrix. This matrix will have the coordinates of the standard basis vectors (1,0) and (0,1) written in terms of the new basis vectors (2,1) and (-1,2). So, the matrix will look like:

| 2 -1 |
| 1 2 |

Multiplying this matrix by the coordinates of our point (3,4) in the standard basis, we get:

| 2 -1 | | 3 | | 5 |
| 1 2 | * | 4 | = | 6 |

This means that in the new basis, our point has coordinates (5,6). This intuitively makes sense because we can see that the new basis vectors (2,1) and (-1,2) are longer and at different angles compared to the standard basis vectors. So, the coordinates of our point in the new basis will be different.

In general, the change of basis matrix allows us to translate coordinates from one basis to another, by representing the new basis vectors in terms of the old basis vectors. This is important in linear algebra because it allows us to work with different bases and still perform calculations using matrix operations.
 

1. What is a change of basis matrix?

A change of basis matrix is a mathematical tool used to convert coordinates from one basis to another. It represents the transformation between two coordinate systems.

2. How is a change of basis matrix calculated?

To calculate a change of basis matrix, you first need to determine the basis vectors for both the original and new coordinate systems. Then, the change of basis matrix can be formed by placing the basis vectors of the new system as columns in a matrix, with each column representing the coordinates of a respective basis vector in the original system.

3. What is the significance of a change of basis matrix?

A change of basis matrix is important in linear algebra and other fields of mathematics because it allows for the representation of vectors and transformations in different coordinate systems. It is also used in many applications, such as computer graphics and quantum mechanics.

4. Can a change of basis matrix be non-square?

Yes, a change of basis matrix can be non-square. In fact, it is common for a change of basis matrix to be a rectangular matrix, as it represents a transformation between two coordinate systems with potentially different dimensions.

5. How is a change of basis matrix applied to a vector?

To apply a change of basis matrix to a vector, you simply multiply the vector by the change of basis matrix. This will result in a new vector with coordinates expressed in the new coordinate system.

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