Orthogonality means $\langle u,v\rangle=0$. For $u\ne0$, the projection of $v$ onto its line is $u\langle u,v\rangle/\|u\|^2$ under the convention conjugating the first slot. For an orthonormal basis matrix $Q$ of a subspace $W$, $P_Wv=QQ^*v$.
The residual $v-P_Wv$ lies in $W^\perp$, giving the unique closest point in $W$. This leads directly to least squares, Gram–Schmidt, and QR.
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- Inner products in coordinate spacesIn real coordinates the standard inner product is $\langle x,y\rangle=x^Ty$. In complex coordinates use $\langle...
- Norm and lengthThe norm induced by an inner product is $\ x\ =\sqrt{\langle x,x\rangle}$. In standard coordinates it is Euclidean...
- OrthogonalityVectors are orthogonal if their inner product is zero. In real Euclidean space this means perpendicular directions...
- ProjectionA projection is a linear operator $P$ with $P^2=P$. It acts as the identity on its image and as zero on its kernel....
- Least squaresLeast squares solves $\min x\ Ax-b\ 2$. A minimizer always exists in finite dimensions, and its fitted vector $A\hat...
- Inner productsAn inner product adds metric geometry to a real or complex vector space. This graph uses the convention $\langle...
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