Vectors are orthogonal if their inner product is zero. In real Euclidean space this means perpendicular directions when both are nonzero. The zero vector is orthogonal to every vector. Pairwise orthogonal nonzero vectors form an independent list; orthogonality alone allows zero vectors.
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- Definition of angle between vectorsFor nonzero real vectors, define $\theta\in[0,\pi]$ by $\cos\theta=\langle x,y\rangle/(\ x\ \ y\ )$. Cauchy–Schwarz...
- Definition of orthogonal vectorsVectors $x,y$ in an inner-product space are orthogonal, written $x\perp y$, when $\langle x,y\rangle=0$. Example:...
- Definition of parallel vectorsNonzero vectors are parallel if one is a nonzero scalar multiple of the other. In real space a positive scalar gives...
- Pythagorean equality characterizes orthogonality over the realsIn a real inner-product space, $x\perp y$ iff $\ x+y\ ^2=\ x\ ^2+\ y\ ^2$, since the difference is $2\langle...
- Definition of a vector being orthogonal to a subspaceA vector $x$ is orthogonal to a subspace $W$ if $\langle w,x\rangle=0$ for every $w\in W$. Conjugate symmetry makes...
- Definition of orthogonal complement of a subspaceThe orthogonal complement of $W\subseteq V$ is $W^\perp=\{x\in V:\langle w,x\rangle=0\text{ for every }w\in W\}$. It...
- The orthogonal complement of a subspace is a subspace.If $x,y\in W^\perp$, then for every $w\in W$, $\langle w,ax+by\rangle=a\langle w,x\rangle+b\langle w,y\rangle=0$....
- The direct sum of a subspace and its orthogonal complement is the whole space.In a finite-dimensional inner-product space, $V=W\oplus W^\perp$. Choose an orthonormal basis of $W$ and extend it to...
- A vector is in the orthogonal complement of a subspace if and only if it is orthogonal to every vector in a basis of the subspace.If $w 1,\ldots,w k$ span $W$, then $x\in W^\perp$ iff $\langle w i,x\rangle=0$ for every $i$. One direction is...
- Null spaces are orthogonal complements of the corresponding row or column spacesFor $A\in\mathbb C^{m\times n}$, $\ker A=(\operatorname{range}A^ )^\perp\subseteq\mathbb C^n$ and $\ker A^...
- Definition of orthogonal set of vectorsA list is orthogonal if every pair of distinct entries has inner product zero. An orthogonal list may include zero...
- Definition of orthonormal set of vectorsAn orthonormal list satisfies $\langle q i,q j\rangle=\delta {ij}$: distinct vectors are orthogonal and every vector...
- An orthogonal set of nonzero vectors is linearly independent.Suppose nonzero $u i$ are pairwise orthogonal and $\sum i c iu i=0$. Taking the inner product with $u j$ in the first...
- Definition of orthogonal basis of a (sub)spaceAn orthogonal basis is a basis whose distinct vectors are orthogonal. Its vectors are nonzero by independence, but...
- Definition of orthonormal basis of a (sub)spaceAn orthonormal basis is an orthogonal basis of unit vectors. If $q 1,\ldots,q n$ is such a basis, $v=\sum jq j\langle...
- A square real matrix has orthonormal columns iff its inverse is its transposeFor a square real matrix $Q$, orthonormal columns are equivalent to $Q^TQ=I$, which implies $Q^{-1}=Q^T$. For...
- A matrix with real entries and orthonormal columns preserves norms.If real $Q$ has orthonormal columns, $\ Qx\ ^2=x^TQ^TQx=x^Tx=\ x\ ^2$. Thus it preserves lengths of domain vectors,...
- A matrix with real entries and orthonormal columns preserves dot products.If real $Q^TQ=I$, then $\langle Qx,Qy\rangle=x^TQ^TQy=x^Ty=\langle x,y\rangle$. Thus dot products and angles are...
- Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.For an orthogonal basis $(u i)$, $[v] B$ has entries $c i=\langle u i,v\rangle/\ u i\ ^2$. If the basis is...
- A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.For finite-dimensional $W$ in an inner-product space, every $v$ has a unique decomposition $v=w+z$ with $w\in W$ and...
- Description of the Gram-Schmidt processGiven independent $v 1,\ldots,v k$, set $u 1=v 1$ and $u j=v j-\sum {i
- The Gram-Schmidt process converts a linearly independent set into an orthogonal set.Gram–Schmidt turns an independent list into an orthogonal nonzero list with the same span at every prefix....
- Definition of Gram-Schmidt processGram–Schmidt orthogonalizes an independent list: $u j=v j-\sum {i
- Definition of orthogonal subspacesSubspaces $U,W$ of an inner-product space are orthogonal if $\langle u,w\rangle=0$ for every $u\in U,w\in W$. Their...
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