How to Construct an Orthonormal Basis for a 2D Subspace in Linear Algebra?

In summary: Generalization_to_non-orthogonal_setsSo in summary, to construct an orthonormal basis for the space W, we can use the Gram-Schmidt process on the image of V under the map A, as long as the inner product is defined within the space.
  • #1
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I have two n-vectors [itex]e_1, e_2[/itex] which span a 2D subspace of [itex]R^n[/itex]:
[tex]
V = span\{e_1,e_2\}
[/tex]
The vectors [itex]e_1,e_2[/itex] are not necessarily orthogonal (but they are not parallel so we know its a 2D and not a 1D subspace). Now I also have a linear map:
[tex]
f: V \rightarrow W \\
f(v) = A v
[/tex]
where [itex]A[/itex] is a given [itex]n \times n[/itex] invertible matrix.

My question is: how would I construct an orthonormal basis for the space [itex]W[/itex]?

My thinking is to perform a QR decomposition on the [itex]n \times 2[/itex] matrix
[tex]
\left(
\begin{array}{cc}
A e_1 & A e_2
\end{array}
\right)
[/tex]
and then the columns of [itex]Q[/itex] will be an orthonormal basis for [itex]W[/itex]. Is this a correct solution? I'm not entirely sure since [itex]e_1,e_2[/itex] are not orthonormal.
 
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How do you define ##W##? Is ##W## the image of ##V## under the map ##A## (as a linear operator in ##R^n##)? In this case ##W## is two dimensional if ##A e_1## and ##A e_2## are linearly independent, and is one dimensional otherwise.

In the case where ##W## is two dimensional, you know ##A e_1## and ##A e_2## form a basis. Use the Gram-Schmidt process to find orthonormal basis.
 
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  • #3
Lucas SV said:
How do you define ##W##? Is ##W## the image of ##V## under the map ##A## (as a linear operator in ##R^n##)? In this case ##W## is two dimensional if ##A e_1## and ##A e_2## are linearly independent, and is one dimensional otherwise.

In the case where ##W## is two dimensional, you know ##A e_1## and ##A e_2## form a basis. Use the Gram-Schmidt process to find orthonormal basis.
Yes, we can think of [itex]W[/itex] as the image of [itex]V[/itex] under the map [itex]A[/itex]
 
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Related to How to Construct an Orthonormal Basis for a 2D Subspace in Linear Algebra?

1. What is a change of basis?

A change of basis is a mathematical operation in which the coordinates of a vector or matrix are represented in terms of a different set of basis vectors. This allows for the transformation of a vector or matrix from one coordinate system to another.

2. Why is change of basis important in mathematics?

Change of basis is important in mathematics because it allows for the simplification and analysis of complex problems by transforming them into a more convenient coordinate system. It also plays a crucial role in linear algebra, differential equations, and multivariable calculus.

3. What are the common methods for performing a change of basis?

The most common methods for performing a change of basis include the use of matrices, the Gram-Schmidt process, and the eigenvalue decomposition method. Each method has its own advantages and is suitable for different types of problems.

4. How is change of basis used in real-world applications?

Change of basis is used in a variety of real-world applications, such as image and signal processing, computer graphics, and machine learning. It is also essential in physical sciences, engineering, and economics for modeling and solving complex problems.

5. What are some common challenges when working with change of basis?

Some common challenges when working with change of basis include understanding the concept and its application, choosing the appropriate method for a specific problem, and dealing with numerical errors and round-off effects. It also requires a solid understanding of linear algebra and mathematical concepts.

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