Orthogonal complement

Transcription

1 Orthogonal complement Aim lecture: Inner products give a special way of constructing vector space complements. As usual, in this lecture F = R or C. We also let V be an F-space equipped with an inner product ( ). Defn Let S V. We define the orthogonal complement to S to be S = {v V v S} = w S ker(w ) Hence S is a subspace orthogonal to S & in particular, is closed under addition. Proof. Clear. E.g. This concept is easily understood in R 3 Daniel Chan (UNSW) Lecture 33: Orthogonal complements & projections Semester / 10

2 Orthogonal complements of spans Lemma Let S V. Then Span(S) = S Proof. w S v w for all v S w v for all v S S ker(w ) Span(S) ker(w ) w Span(S) This completes the proof. E.g. The orthogonal complement to S = Span((1, 1, 0) T, (0, 1, 1) T ) is Daniel Chan (UNSW) Lecture 33: Orthogonal complements & projections Semester / 10

3 Orthogonal (internal) direct sums Prop-Defn Let W 1,..., W r V be mutually orthogonal subspaces i.e. W i W j whenever i j. Then the sum r i=1 W i is direct & we say the internal direct sum i W i is orthogonal. Proof. The lemma ensures that W r is orthogonal to W <r = r 1 i=1 W i so by induction, it suffices to show that any w W r W <r W r Wr must be 0. But w w so (w w) = 0 & w = 0. This completes the proof. E.g. Daniel Chan (UNSW) Lecture 33: Orthogonal complements & projections Semester / 10

4 Vector space complement Prop Let W V & dim W < then W + W = V so V = W W. Proof. We prove this here only in the case where dim V <. The propn on orthog direct sums ensures the sum W + W is direct. Pick a basis w 1,..., w r W for W. For i = 1,..., r we have l i = (w i ) L(V, F) so we may form the r 1-matrix T whose i-th entry is l i. Note T : V F r : v (l 1 (v),..., l r (v)) T. By the lemma W = i ker l i = ker T. Since im T F r, rank-nullity ensures that dim W = dim V dim im T dim V r = dim V dim W. However, the sum W + W is direct, so we must have dim W + W = dim W + dim W = dim V which ensures V = W + W as desired. Daniel Chan (UNSW) Lecture 33: Orthogonal complements & projections Semester / 10

5 Examples E.g. Let V = C[x] 1 with inner product (f g) = 1 f (t)g(t)dt. Find the 0 orthogonal complement to W = C(1 + ix). Daniel Chan (UNSW) Lecture 33: Orthogonal complements & projections Semester / 10

6 Orthogonal projections, bases Cor-Defn 1 Suppose that V = W W (e.g. when dim W < ). The orthogonal projection onto W is the linear map proj W : V V : ( w w ) ( w 0). 2 For v V we have v proj W v W. 3 If V = W 1... W r is an orthogonal direct sum then Wi particular, if V = W W then (W ) = W. = j i W j. In 4 We say a set S = {w 1,..., w r } V is orthogonal if w i w j for i j. Equivalently, the sum i F w i is an orthogonal direct sum. In particular, S is lin indep in this case. 5 An orthogonal set S V is orthonormal if furthermore, w i = 1 for all i. Proof. 2),4) follow from propns. We prove 3) first noting that W i = j i W j is orthogonal to W i. It thus suffices to show Wi W i so suppose w W i. We may write w = w i + w i with w i W i, w i W i. Then 0 = (w i w) = (w i w i + w i ) = (w i w i) + (w i w i ) = (w i w i ) so w i = 0 & w = w i W i. Daniel Chan (UNSW) Lecture 33: Orthogonal complements & projections Semester / 10

7 Existence of orthonormal bases Theorem Let V be fin dim. Then 1 V is the orthogonal direct sum of 1-dimensional vector spaces. 2 V has an orthonormal basis. Proof. 1) We argue by induction on d = dim V, the cases d = 0, 1 being clear so suppose that d > 1. We may thus pick a non-zero subspace W V e.g. F w for any non-zero w W. Now V = W W & dim W, dim W < d. By induction, each of W & W are orthogonal direct sums of 1-dimensional F-spaces, say W = i W i, W = j V j. Clearly, the subspaces {W i, V j } are still mutually orthog so V is the orthogonal direct sum of them. 2) By 1), it suffices to find an orthonormal basis for a 1-dim F-space F v. Just pick v v. Daniel Chan (UNSW) Lecture 33: Orthogonal complements & projections Semester / 10

8 Orthogonal projection formula Prop 1 Given a 1-dim F-space W = F w we have V = W W & proj W v = (w v) w 2 w for any v V. 2 Suppose that V = W i Wi for i = 1,..., r so we have orthog projn maps proj Wi. Suppose further that the W i are mutually orthog so we may consider the orthogonal direct sum W = W i. Then V = W W & proj W = i proj W i. 3 ( Fourier decomposition ) In particular, if W is spanned by the orthog set {w 1,..., w r }, then r (w i v) proj W v = w i 2 w i. Rem This gives a proof of the propn on vector space complements in general. Proof. Note 3) follows immediately from 1) & 2) i=1 Daniel Chan (UNSW) Lecture 33: Orthogonal complements & projections Semester / 10

9 Proof propn 1) We need only show v = v (w v) w 2 w W for then we see v = (w v) w 2 w + v W + W. But (w v ) = (w v (w v) (w v) w) = (w v) w 2 w 2 (w w) = 0 2) As above V = W W follows from showing v i proj W i v W. In this case we may write V = W 1... W r W & writing v = (w 1,..., w r, w ) T for w i W i, w W we see w 1 w proj W v =. = = proj Wi v. w r 0 w r i Daniel Chan (UNSW) Lecture 33: Orthogonal complements & projections Semester / 10

10 Example E.g. Consider the orthonormal basis f 1 (x) = 1, f 2 (x) = 2 3x 3 for W = R[x] 1 (wrt (f g) = 1 0 f (t)g(t)dt). Find proj W x 2. Daniel Chan (UNSW) Lecture 33: Orthogonal complements & projections Semester / 10