Change of basis in R^n and dimension is <n

In summary, the conversation discusses the process of changing basis for a subspace V in \mathbb{R}^{4} with basis vectors \mathbf{v_{1}}, \mathbf{v_{2}}, and \mathbf{v_{3}}. The dimension of V is 3, but it is in \mathbb{R}^{4}. The conversation explores different methods of changing basis and clarifies that the number of components in the coordinate system is determined by the dimension of the subspace, not the dimension of the space it is in.
  • #1
IniquiTrance
190
0
Suppose I have a basis for a subspace V in [tex]\mathbb{R}^{4}[/tex]:

[tex]\mathbf{v_{1}}=[1, 3, 5, 7]^{T}[/tex]
[tex]\mathbf{v_{2}}=[2, 4, 6, 8]^{T}[/tex]
[tex]\mathbf{v_{3}}=[3, 3, 4, 4]^{T}[/tex]

V has dimension 3, but is in [tex]\mathbb{R}^{4}[/tex]. How would one switch basis for this subspace, when you can't use an invertible transition matrix?

Thanks!
 
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  • #2
I don't really understand the question very well. "Changing basis" is a process of converting the representation of vectors in a space from one basis to another. But here you have two different spaces. It's kind of like asking "how do you change basis between [itex] \mathbb{R}^{3} [/itex] and [itex] \mathbb{R}^{2} [/itex]?"...well, you can't.
 
  • #3
Hmm, I guess I'm asking how one would switch to another another basis that spans V. Given co-ordinates in terms of v1, v2, v3, how would one find co-ordinates with respect to a new spanning set of vectors?

If V spanned R4, then I could just use a transition matrix. It's the fact that V is dimension 3, but each basis vector has 4 co-ordinates, that is confusing me.
 
  • #4
Well, if they gave you coordinates in terms of v1, v2, and v3, then you'd have a vector with 3 components, right? And any other basis for the same space would also have 3 components, because the dimension of the subspace is 3. So then you have your usual 3x3 thing for changing basis between those.
 
  • #5
But how can you use a square matrix when the co-ordinate vectors have 4 components? This is what's confusing me...
 
  • #6
You don't have four components, because instead of writing out vectors of v in terms of the standard basis in R4, you write them out in coordinates in terms of v1, v2 and v3. Then you have three coordinates as required. For example, the vector (1,3,1) in this coordinate system would be 1*v1+3v2+1v3=(10,18,27,35)
 
  • #7
Office_Shredder said:
You don't have four components, because instead of writing out vectors of v in terms of the standard basis in R4, you write them out in coordinates in terms of v1, v2 and v3. Then you have three coordinates as required. For example, the vector (1,3,1) in this coordinate system would be 1*v1+3v2+1v3=(10,18,27,35)

FFFFFFFFFFFUUUUUUUUUUUUUUUUU-

Great explanation!

Thanks for the responses!
 

What is change of basis in R^n?

Change of basis in R^n refers to the process of representing a vector or a set of vectors in a different coordinate system or basis. It involves finding the linear combination of the basis vectors that can represent the original vector(s).

How is change of basis related to dimension in R^n?

Change of basis is closely related to dimension in R^n. The dimension of a vector space is the number of linearly independent basis vectors that can represent any vector in the space. When changing the basis, the dimension of the vector space remains the same, but the basis vectors themselves change.

What are the steps to perform change of basis in R^n?

The steps to perform change of basis in R^n are:1. Determine the current basis vectors and their coordinates in the new basis.2. Construct a transformation matrix using the coordinates of the current basis vectors.3. Multiply the transformation matrix by the vector(s) to be transformed.4. The resulting vector(s) will be the coordinates of the original vector(s) in the new basis.

What is the purpose of performing change of basis in R^n?

The purpose of performing change of basis in R^n is to make calculations and representations of vectors easier. By changing the basis, we can work with simpler and more convenient basis vectors, which can make solving problems and understanding vector operations easier.

Can change of basis in R^n lead to a loss of information?

No, change of basis in R^n does not lead to a loss of information. The original vector(s) can always be reconstructed from the new basis and its coordinates using the inverse transformation matrix. Therefore, no information is lost in the process of changing the basis.

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