Comparing Basis Vectors in Linear Spaces: X and Y

In summary, the conversation discusses finding an algebraic argument to show that X(X'X)^(-1)X' = Y(Y'Y)^(-1)Y', where X and Y are matrices with columns representing vectors that form bases for linear space V1. It is suggested to use the fact that (AB)^(-1) = B^(-1)A^(-1) and to consider the dimensions of X and Y. It is also mentioned that X(X'X)^(-1)X' is a projection operator and that if all columns in X are linearly independent, then X must be non-singular and have an inverse. Finally, it is noted that A and B span V1, a subspace of
  • #1
julie94
18
0
Hi everyone,

I am working on the following problem.

Suppose the set of vectors X1,..,Xk is a basis for linear space V1.
Suppose the set of vectors Y1,..,Yk is also a basis for linear space
V1.
Clearly the linear space spanned by the Xs equals the linear space
spanned by the Ys.

Set
X=[X1: X2 :...: Xk]
Y=[Y1: Y2 :...: Yk]

Construct an algebraic argument to show that
X(X'X)^(-1)X'=Y(Y'Y)^(-1)Y'

This is the idea I have:
X=PYP^{-1}
where P changes the basis from Y to X.

Is this the right avenue?Thanks in advance.
 
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  • #2
Didn't we just have this? Use the fact that (AB)-1= B-1A-1.
 
  • #3
Thanks for the help.

There is a problem with the dimensions.

X is n*k

because each of X1,..,Xk is n*1

So I was going to write (using your help)
X(X'X)^{-1}X'= X(X^{-1}X'^{-1})X'
But X^{-1} does not exist
 
Last edited:
  • #4
I think I need to use the fact that X(X'X)^(-1)X' is a projection operator somehow.
 
  • #5
If all columns in X are linearly independent- and you said they formed a basis for V1- then X must be non-singular and have an inverse!
 
  • #6
X is not a square matrix
 
  • #7
Ah! so A and B span V1, a subspace of V!
 
  • #8
yes

any ideas?
 

FAQ: Comparing Basis Vectors in Linear Spaces: X and Y

1. What is the purpose of comparing basis vectors in linear spaces: X and Y?

The purpose of comparing basis vectors in linear spaces: X and Y is to understand the relationship between the two sets of basis vectors and how they can be used to represent the same vector space. This comparison helps in understanding the similarities and differences between the two sets of basis vectors and their corresponding linear transformations.

2. How do you compare basis vectors in linear spaces: X and Y?

Basis vectors in linear spaces: X and Y can be compared by looking at their properties such as magnitude, direction, and orthogonality. One can also compare the span of the basis vectors and their linear combinations to understand how they represent the same vector space. Additionally, one can compare the transformation matrices associated with the basis vectors to see how they map vectors from one space to another.

3. What is the importance of comparing basis vectors in linear spaces: X and Y?

Comparing basis vectors in linear spaces: X and Y is important in understanding the structure and properties of vector spaces. It helps in visualizing how different sets of basis vectors can represent the same vector space and how they can be used to perform linear transformations. This comparison also aids in understanding the concept of change of basis and its applications in various fields of science and engineering.

4. Can basis vectors in linear spaces: X and Y be equivalent?

Yes, basis vectors in linear spaces: X and Y can be equivalent if they span the same vector space and their linear combinations can represent the same set of vectors. However, they may not be identical as they can have different magnitudes, directions, and orthogonalities. It is important to compare their properties to determine their equivalence.

5. What are the limitations of comparing basis vectors in linear spaces: X and Y?

One limitation of comparing basis vectors in linear spaces: X and Y is that it may not always be possible to compare them directly. This is because the basis vectors may have different dimensions or they may represent different vector spaces. In such cases, it is important to use appropriate techniques, such as projection or transformation, to compare the basis vectors and understand their relationship.

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