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2 - The Big Picture of Linear Algebra

2020-05-24 19:20 CST
2020-05-24 22:11 CST
CC BY-NC 4.0

Orthogonal

If $Ax = 0$, then $x$ is orthogonal to every row of $A$. (In geometry, perpendicular.)

  • Every $x$ in the nullspace of $A$ is orthogonal to the row space of $A$.
  • Every $y$ in the nullspace of $A^T$ is orthogonal to the column space of $A$.

That is to say:

  • $\textsf{N}(A) \perp \textsf{C}(A^T)$
  • $\textsf{N}(A^T) \perp \textsf{C}(A)$

Two pairs of orthogonal subspaces.

Picture

Elimination on $Ax = b$

E.g. $A = \begin{bmatrix} 2 & 3 \\ 4 & 7 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 2 & 3 \\ 0 & 1 \end{bmatrix} = LU$

$L$ and $U$ are lower and upper triangular matrixs.

If rows are exchanged then $PA = LU$, then $P$ is a permutation.

To solve the equation $Ax = b$, we need to factor $A = LU$.