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 $R^n$:
$$V = span\{e_1,e_2\}$$
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:
$$f: V \rightarrow W \\ f(v) = A v$$
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
$$\left( \begin{array}{cc} A e_1 & A e_2 \end{array} \right)$$
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