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

[vibe3]

I have two n-vectors e_1, e_2 which span a 2D subspace of R^n:
<br /> V = span\{e_1,e_2\}<br />
The vectors e_1,e_2 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:
<br /> f: V \rightarrow W \\<br /> f(v) = A v<br />
where A is a given n \times n invertible matrix.

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

My thinking is to perform a QR decomposition on the n \times 2 matrix
<br /> \left(<br /> \begin{array}{cc}<br /> A e_1 &amp; A e_2<br /> \end{array}<br /> \right)<br />
and then the columns of Q will be an orthonormal basis for W. Is this a correct solution? I'm not entirely sure since e_1,e_2 are not orthonormal.

 

[Lucas SV]

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.

 

[vibe3]

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 W as the image of V under the map A

 

[chiro]

Hey vibe3.

If an inner product is defined within the space (and is consistent) then I'd try looking at something like this:

https://en.wikipedia.org/wiki/Gram–Schmidt_process