6.2 Orthogonal Sets Math 2331 { Linear Algebra 6.2 Orthogonal Sets Jiwen He Department of Mathematics, University of Houston [email protected] math.uh.edu/∼jiwenhe/math2331 Jiwen He, University of Houston Math 2331, Linear Algebra 1 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix 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 2331, Linear Algebra 2 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Sets Orthogonal Sets n A set of vectors fu1; u2;:::; upg in R is called an orthogonal set if ui · uj = 0 whenever i 6= j. Example 82 3 2 3 2 39 < 1 1 0 = Is 4 −1 5 ; 4 1 5 ; 4 0 5 an orthogonal set? : 0 0 1 ; Solution: Label the vectors u1; u2; and u3 respectively. Then u1 · u2 = u1 · u3 = u2 · u3 = Therefore, fu1; u2; u3g is an orthogonal set. Jiwen He, University of Houston Math 2331, Linear Algebra 3 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Sets: Theorem Theorem (4) Suppose S = fu1; u2;:::; upg is an orthogonal set of nonzero n vectors in R and W =spanfu1; u2;:::; upg. Then S is a linearly independent set and is therefore a basis for W . Partial Proof: Suppose c1u1 + c2u2 + ··· + cpup = 0 (c1u1 + c2u2 + ··· + cpup) · = 0· (c1u1) · u1 + (c2u2) · u1 + ··· + (cpup) · u1 = 0 c1 (u1 · u1) + c2 (u2 · u1) + ··· + cp (up · u1) = 0 c1 (u1 · u1) = 0 Since u1 6= 0, u1 · u1 > 0 which means c1 = : In a similar manner, c2,:::,cp can be shown to by all 0. So S is a linearly independent set. Jiwen He, University of Houston Math 2331, Linear Algebra 4 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Basis Orthogonal Basis: Example An orthogonal basis for a subspace W of Rn is a basis for W that is also an orthogonal set. Example Suppose S = fu1; u2;:::; upg is an orthogonal basis for a subspace n W of R and suppose y is in W . Find c1,:::,cp so that y =c1u1 + c2u2 + ··· + cpup. Solution: y· = (c1u1 + c2u2 + ··· + cpup) · y · u1= (c1u1 + c2u2 + ··· + cpup) · u1 y · u1=c1 (u1 · u1) + c2 (u2 · u1) + ··· + cp (up · u1) y · u =c (u · u ) =) c = y·u1 1 1 1 1 1 u1·u1 Similarly, c2 = ,. , cp = Jiwen He, University of Houston Math 2331, Linear Algebra 5 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Basis: Theorem Theorem (5) Let fu1; u2;:::; upg be an orthogonal basis for a subspace W of Rn: Then each y in W has a unique representation as a linear combination of u1; u2;:::; up. In fact, if y =c1u1 + c2u2 + ··· + cpup then y · uj cj = (j = 1;:::; p) uj · uj Jiwen He, University of Houston Math 2331, Linear Algebra 6 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Basis: Example Example 2 3 3 Express y = 4 7 5 as a linear combination of the orthogonal basis 4 82 3 2 3 2 39 < 1 1 0 = 4 −1 5 ; 4 1 5 ; 4 0 5 . : 0 0 1 ; Solution: y · u y · u y · u 1 = 2 = 3 = u1 · u1 u2 · u2 u3 · u3 Hence y = u1 + u2 + u3 Jiwen He, University of Houston Math 2331, Linear Algebra 7 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Projections For a nonzero vector u in Rn, suppose we want to write y in Rn as the the following y = (multiple of u) + (multiple a vector ? to u) (y − αu) · u =0 =) y · u − α (u · u) =0 =) α = y·u by= u·u u (orthogonal projection of y onto u) y·u z = y− u·u u (component of y orthogonal to u) Jiwen He, University of Houston Math 2331, Linear Algebra 8 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Projections: Example Example −8 3 Let y = and u = . Compute the distance from y to 4 1 the line through 0 and u. Solution: y · u y= u = b u · u Distance from y to the line through 0 and u = distance from by to y = kby − yk = Jiwen He, University of Houston Math 2331, Linear Algebra 9 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthonormal Sets Orthonormal Sets n A set of vectors fu1; u2;:::; upg in R is called an orthonormal set if it is an orthogonal set of unit vectors. Orthonormal Basis If W =spanfu1; u2;:::; upg, then fu1; u2;:::; upg is an orthonormal basis for W : p p Recall that v is a unit vector if kvk = v · v = vT v = 1. Jiwen He, University of Houston Math 2331, Linear Algebra 10 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthonormal Matrix: Example Example Suppose U = [u1 u2 u3] where fu1; u2; u3g is an orthonormal set. 2 3 2 3 2 T 3 u1 T T 6 7 6 7 U U = 4 u2 5 [u1 u2 u3] = 6 7 = 6 7 T 4 5 4 5 u3 Orthogonal Matrix It can be shown that UUT = I : So U−1 = UT (such a matrix is called an orthogonal matrix). Jiwen He, University of Houston Math 2331, Linear Algebra 11 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthonormal Matrix: Theorems Theorem (6) An m × n matrix U has orthonormal columns if and only if UT U = I : Theorem (7) Let U be an m × n matrix with orthonormal columns, and let x and y be in Rn. Then a. kUxk = kxk b.( Ux) · (Uy) = x · y c.( Ux) · (Uy) = 0 if and only if x · y = 0. Proof of part b: (Ux) · (Uy) = Jiwen He, University of Houston Math 2331, Linear Algebra 12 / 12.