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Luc Bijl 2025-08-28 17:12:46 +02:00
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docs/mathematics/linear-algebra/orthogonality.md
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@ -46,7 +46,7 @@ for all $\mathbf{v} \in S$, and hence $\mathbf{u}_1 + \mathbf{u}_2 \in S^\perp$.
### Fundamental subspaces
Let $V$ be an Euclidean inner product space $V = \mathbb{R}^n$ with its inner product defined by the [scalar product](../inner-product-spaces/#euclidean-inner-product-spaces). With this definition of the inner product on $V$ the following theorem may be posed.
Let $V$ be an Euclidean inner product space $V = \mathbb{R}^n$ with its inner product defined by the [scalar product](inner-product-spaces.md#euclidean-inner-product-spaces). With this definition of the inner product on $V$ the following theorem may be posed.
> *Theorem 1*: let $A$ be an $m \times n$ matrix, then
>
@ -118,7 +118,7 @@ Known as the fundamental theorem of linear algebra. Which can be used to prove t
S^\perp = R(X^T)^\perp = N(X),
$$
from the [rank nullity theorem](../vector-spaces/#rank-and-nullity) it follows that
from the [rank nullity theorem](vector-spaces.md#rank-and-nullity) it follows that
$$
\dim S^\perp = \dim N(X) = n - r.
@ -464,4 +464,4 @@ $$
\mathbf{\hat x} = (A^T A)^{-1} A^T \mathbf{b},
$$
is the unique solution of the normal equations for $A$ nonsingular and consequently, the unique least squares solution of the system $A \mathbf{x} = \mathbf{b}$.
is the unique solution of the normal equations for $A$ nonsingular and consequently, the unique least squares solution of the system $A \mathbf{x} = \mathbf{b}$.