Orthogonal Complements Notation F denotes either R or C. V denotes an inner product space over F. If U is a subset of V, then U? is a subspace of V. f0g? = V. V? = f0g. If U is a subset of V, then If U is a line in R3 containing the origin, U \ U? ⊂ f0g. ? then U is the plane containing the If U and W are subsets of V origin that is perpendicular to U. and U ⊂ W, then W? ⊂ U?. If U is a plane in R3 containing the origin, then U? is the line containing the origin that is perpendicular to U. Basic properties of orthogonal complement Orthogonal Complements Definition: orthogonal complement, U? If U is a subset of V, then the orthogonal complement of U, denoted U?, is the set of all vectors in V that are orthogonal to every vector in U: U? = fv 2 V : hv; ui = 0 for every u 2 Ug: If U is a subset of V, then U? is a subspace of V. f0g? = V. V? = f0g. If U is a subset of V, then U \ U? ⊂ f0g. If U and W are subsets of V and U ⊂ W, then W? ⊂ U?. Basic properties of orthogonal complement If U is a plane in R3 containing the origin, then U? is the line containing the origin that is perpendicular to U. Orthogonal Complements Definition: orthogonal complement, U? If U is a subset of V, then the orthogonal complement of U, denoted U?, is the set of all vectors in V that are orthogonal to every vector in U: U? = fv 2 V : hv; ui = 0 for every u 2 Ug: If U is a line in R3 containing the origin, then U? is the plane containing the origin that is perpendicular to U. If U is a subset of V, then U? is a subspace of V. f0g? = V. V? = f0g. If U is a subset of V, then U \ U? ⊂ f0g. If U and W are subsets of V and U ⊂ W, then W? ⊂ U?. Basic properties of orthogonal complement Orthogonal Complements Definition: orthogonal complement, U? If U is a subset of V, then the orthogonal complement of U, denoted U?, is the set of all vectors in V that are orthogonal to every vector in U: U? = fv 2 V : hv; ui = 0 for every u 2 Ug: If U is a line in R3 containing the origin, then U? is the plane containing the origin that is perpendicular to U. If U is a plane in R3 containing the origin, then U? is the line containing the origin that is perpendicular to U. If U is a subset of V, then U? is a subspace of V. f0g? = V. V? = f0g. If U is a subset of V, then U \ U? ⊂ f0g. If U and W are subsets of V and U ⊂ W, then W? ⊂ U?. Basic properties of orthogonal complement Orthogonal Complements Definition: orthogonal complement, U? If U is a subset of V, then the orthogonal complement of U, denoted U?, is the set of all vectors in V that are orthogonal to every vector in U: U? = fv 2 V : hv; ui = 0 for every u 2 Ug: If U is a line in R3 containing the origin, then U? is the plane containing the origin that is perpendicular to U. If U is a plane in R3 containing the origin, then U? is the line containing the origin that is perpendicular to U. f0g? = V. V? = f0g. If U is a subset of V, then U \ U? ⊂ f0g. If U and W are subsets of V and U ⊂ W, then W? ⊂ U?. Orthogonal Complements Definition: orthogonal complement, U? Basic properties of orthogonal complement If U is a subset of V, then the orthogonal ? complement of U, denoted U , is the set of If U is a subset of V, then U? is all vectors in V that are orthogonal to every a subspace of V. vector in U: U? = fv 2 V : hv; ui = 0 for every u 2 Ug: If U is a line in R3 containing the origin, then U? is the plane containing the origin that is perpendicular to U. If U is a plane in R3 containing the origin, then U? is the line containing the origin that is perpendicular to U. V? = f0g. If U is a subset of V, then U \ U? ⊂ f0g. If U and W are subsets of V and U ⊂ W, then W? ⊂ U?. Orthogonal Complements Definition: orthogonal complement, U? Basic properties of orthogonal complement If U is a subset of V, then the orthogonal ? complement of U, denoted U , is the set of If U is a subset of V, then U? is all vectors in V that are orthogonal to every a subspace of V. vector in U: f0g? = V. U? = fv 2 V : hv; ui = 0 for every u 2 Ug: If U is a line in R3 containing the origin, then U? is the plane containing the origin that is perpendicular to U. If U is a plane in R3 containing the origin, then U? is the line containing the origin that is perpendicular to U. If U is a subset of V, then U \ U? ⊂ f0g. If U and W are subsets of V and U ⊂ W, then W? ⊂ U?. Orthogonal Complements Definition: orthogonal complement, U? Basic properties of orthogonal complement If U is a subset of V, then the orthogonal ? complement of U, denoted U , is the set of If U is a subset of V, then U? is all vectors in V that are orthogonal to every a subspace of V. vector in U: f0g? = V. ? U = fv 2 V : hv; ui = 0 for every u 2 Ug: V? = f0g. If U is a line in R3 containing the origin, then U? is the plane containing the origin that is perpendicular to U. If U is a plane in R3 containing the origin, then U? is the line containing the origin that is perpendicular to U. If U and W are subsets of V and U ⊂ W, then W? ⊂ U?. Orthogonal Complements Definition: orthogonal complement, U? Basic properties of orthogonal complement If U is a subset of V, then the orthogonal ? complement of U, denoted U , is the set of If U is a subset of V, then U? is all vectors in V that are orthogonal to every a subspace of V. vector in U: f0g? = V. ? U = fv 2 V : hv; ui = 0 for every u 2 Ug: V? = f0g. If U is a subset of V, then If U is a line in R3 containing the origin, U \ U? ⊂ f0g. then U? is the plane containing the origin that is perpendicular to U. If U is a plane in R3 containing the origin, then U? is the line containing the origin that is perpendicular to U. Orthogonal Complements Definition: orthogonal complement, U? Basic properties of orthogonal complement If U is a subset of V, then the orthogonal ? complement of U, denoted U , is the set of If U is a subset of V, then U? is all vectors in V that are orthogonal to every a subspace of V. vector in U: f0g? = V. ? U = fv 2 V : hv; ui = 0 for every u 2 Ug: V? = f0g. If U is a subset of V, then If U is a line in R3 containing the origin, U \ U? ⊂ f0g. ? then U is the plane containing the If U and W are subsets of V origin that is perpendicular to U. and U ⊂ W, then W? ⊂ U?. If U is a plane in R3 containing the origin, then U? is the line containing the origin that is perpendicular to U. Dimension of the orthogonal complement Suppose V is finite-dimensional and U is a subspace of V. Then dim U? = dim V − dim U: Proof Suppose v 2 V. Let e1;:::; em be an orthonormal basis of U. Let The orthogonal complement of the orthogonal complement u = hv; e1ie1 + ··· + hv; emiem: Then hv − u; eji = hv; eji − hu; eji = 0 for Suppose U is a finite-dimensional sub- ? j = 1;:::; m. Thus v − u 2 U . Now space of V. Then v = u + (v − u); U = (U?)?: showing that v 2 U+U?. Thus V = U+U?. Direct Sum Direct sum of a subspace and its orthogonal complement Suppose U is a finite-dimensional sub- space of V. Then V = U ⊕ U?: Dimension of the orthogonal complement Suppose V is finite-dimensional and U is a subspace of V. Then dim U? = dim V − dim U: The orthogonal complement of the orthogonal complement Then hv − u; eji = hv; eji − hu; eji = 0 for Suppose U is a finite-dimensional sub- ? j = 1;:::; m. Thus v − u 2 U . Now space of V. Then v = u + (v − u); U = (U?)?: showing that v 2 U+U?. Thus V = U+U?. Direct Sum Direct sum of a subspace and its orthogonal complement Suppose U is a finite-dimensional sub- space of V. Then V = U ⊕ U?: Proof Suppose v 2 V. Let e1;:::; em be an orthonormal basis of U. Let u = hv; e1ie1 + ··· + hv; emiem: Dimension of the orthogonal complement Suppose V is finite-dimensional and U is a subspace of V. Then dim U? = dim V − dim U: The orthogonal complement of the orthogonal complement Suppose U is a finite-dimensional sub- ? Thus v − u 2 U . Now space of V. Then v = u + (v − u); U = (U?)?: showing that v 2 U+U?.