In partial $LU$ decomposition of a $n \times n$ matrix A, we have $LU = PA$.

$P$ is a $n \times n$ permutation matrix.

$L$ and $U$ are $n \times n$ lower and upper triangular matrices, respectively.

Since the $U$ is upper triangular the column spaces of $PA$ and $L$ are the same. On other hand, if we apply a full QR factorization to matrix A, we have a $n \times n $ matrix Q and an upper triangular matrices. Thus, the column spaces of $A$ and $Q$ are spanning the same space.

(1) Is it true that the column space of $Q$ and $P^{-1}L$ are the same?(I tested it for particular example but I am not sure it is always true)

(2) What is the relation between the last column of $Q$ and last row $L^{-1}$?

I'm wondering if anyone can please help me. Thank you.


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