# QR decomposition of linearly dependent vectors

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 "