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Suppose $\vec{q_1},\vec{q_2},\vec{q_3}$ are linearly independent column vectors with dimensionality 4.

$ X=[\vec{q_1}, \vec{q_2},(3\vec{q_1}-2\vec{q_2}),\vec{q_3}] $

then $X$ is a $4\times4$ dimensional matrix, it is singular.


Let's perform a QR decomposition, where $\hat{e}_i$ are unit column vectors of dimensionality 4.

$X=QR=[\hat{e}_1,\hat{e}_2,\hat{e}_3?,\hat{e}_4 ] \begin{bmatrix} r_{11}& r_{12} & r_{13} &\cdots \\ 0 & r_{22}& r_{23} & \cdots \\ 0 & 0 & ? & \cdots \\ 0 & 0 & 0 & \cdots \end{bmatrix}$


My first questions is how is $\hat{e}_3$ determinated?

The Gram–Schmidt process doesn't work, because there is no leakage direction, which we can set as basis.

Not sure how Householder method behaves.


Second question is for the Wikipedia:

If A has n linearly independent columns, then the first n columns of Q form an orthonormal basis for the column space of A. More generally, the first k columns of Q form an orthonormal basis for the span of the first k columns of A for any 1 ≤ k ≤ n.1 The fact that any column k of A only depends on the first k columns of Q is responsible for the triangular form of R.1

I don't see why "the first n columns of Q form an orthonormal basis for the column space of A "

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