6.2 Orthogonal Sets Math 233 Linear Algebra 6.2 Orthogonal Sets Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math233 Jiwen He, University of Houston Math 233, Linear Algebra / 2
6.2 Orthogonal Sets Orthogonal Sets: Examples Orthogonal Sets: Theorem Orthogonal Basis: Examples Orthogonal Basis: Theorem Orthogonal Projections Orthonormal Sets Orthonormal Matrix: Examples Orthonormal Matrix: Theorems Jiwen He, University of Houston Math 233, Linear Algebra 2 / 2
Orthogonal Sets 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Sets A set of vectors {u, u 2,..., u p } in R n is called an orthogonal set if u i u j = whenever i j. Example Is,, an orthogonal set? Solution: Label the vectors u, u 2, and u 3 respectively. u u 2 = u u 3 = u 2 u 3 = Therefore, {u, u 2, u 3 } is an orthogonal set. Then Jiwen He, University of Houston Math 233, Linear Algebra 3 / 2
Orthogonal Sets: Theorem Theorem (4) Suppose S = {u, u 2,..., u p } is an orthogonal set of nonzero vectors in R n and W =span{u, u 2,..., u p }. Then S is a linearly independent set and is therefore a basis for W. Partial Proof: Suppose c u + c 2 u 2 + + c p u p = (c u + c 2 u 2 + + c p u p ) = (c u ) u + (c 2 u 2 ) u + + (c p u p ) u = c (u u ) + c 2 (u 2 u ) + + c p (u p u ) = c (u u ) = Since u, u u > which means c =. In a similar manner, c 2,...,c p can be shown to by all. linearly independent set. So S is a Jiwen He, University of Houston Math 233, Linear Algebra 4 / 2
Orthogonal Basis Orthogonal Basis: Example An orthogonal basis for a subspace W of R n is a basis for W that is also an orthogonal set. Example Suppose S = {u, u 2,..., u p } is an orthogonal basis for a subspace W of R n and suppose y is in W. Find c,...,c p so that y =c u + c 2 u 2 + + c p u p. Solution: y = (c u + c 2 u 2 + + c p u p ) y u = (c u + c 2 u 2 + + c p u p ) u y u =c (u u ) + c 2 (u 2 u ) + + c p (u p u ) y u =c (u u ) = c = y u u u Similarly, c 2 =,..., c p = Jiwen He, University of Houston Math 233, Linear Algebra 5 / 2
Orthogonal Basis: Theorem Theorem (5) Let {u, u 2,..., u p } be an orthogonal basis for a subspace W of R n. Then each y in W has a unique representation as a linear combination of u, u 2,..., u p. In fact, if y =c u + c 2 u 2 + + c p u p then c j = y u j u j u j (j =,..., p) Jiwen He, University of Houston Math 233, Linear Algebra 6 / 2
Orthogonal Basis: Example Example Express y = Solution: 3 7 4 as a linear combination of the orthogonal basis,,. Hence y u u u = y u 2 u 2 u 2 = y = u + u 2 + u 3 y u 3 u 3 u 3 = Jiwen He, University of Houston Math 233, Linear Algebra 7 / 2
Orthogonal Projections For a nonzero vector u in R n, suppose we want to write y in R n as the the following y = (multiple of u) + (multiple a vector to u) (y αu) u = = y u α (u u) = = α = ŷ= y u u uu (orthogonal projection of y onto u) z = y y u u uu (component of y orthogonal to u) Jiwen He, University of Houston Math 233, Linear Algebra 8 / 2
Orthogonal Projections: Example Example [ ] [ 8 3 Let y = and u = 4 the line through and u. ]. Compute the distance from y to Solution: ŷ= y u u u u = Distance from y to the line through and u = distance from ŷ to y = ŷ y = Jiwen He, University of Houston Math 233, Linear Algebra 9 / 2
Orthonormal Sets Orthonormal Sets A set of vectors {u, u 2,..., u p } in R n is called an orthonormal set if it is an orthogonal set of unit vectors. Orthonormal Basis If W =span{u, u 2,..., u p }, then {u, u 2,..., u p } is an orthonormal basis for W. Recall that v is a unit vector if v = v v = v T v =. Jiwen He, University of Houston Math 233, Linear Algebra / 2
Orthonormal Matrix: Example Example Suppose U = [u u 2 u 3 ] where {u, u 2, u 3 } is an orthonormal set. U T U = u T u T 2 u T 3 [u u 2 u 3 ] = = Orthogonal Matrix It can be shown that So UU T = I. U = U T (such a matrix is called an orthogonal matrix). Jiwen He, University of Houston Math 233, Linear Algebra / 2
Orthonormal Matrix: Theorems Theorem (6) An m n matrix U has orthonormal columns if and only if U T U = I. Theorem (7) Let U be an m n matrix with orthonormal columns, and let x and y be in R n. Then a. Ux = x b. (Ux) (Uy) = x y c. (Ux) (Uy) = if and only if x y =. Proof of part b: (Ux) (Uy) = Jiwen He, University of Houston Math 233, Linear Algebra 2 / 2