- #1

vibe3

- 46

- 1

[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.