Linear Algebra 6.2 Orthogonal Sets

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 Math 2331, Linear Algebra 2 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Sets Orthogonal Sets A set of vectors {u1 , u2 , . . . , up } in Rn is called an orthogonal set if ui · uj = 0 whenever i 6= j. Example       1 1 0   Is  −1  ,  1  ,  0  an orthogonal set?   0 0 1 Solution: Label the vectors u1 , u2 , and u3 respectively. Then u1 · u2 = u1 · u3 = u2 · u3 = Therefore, {u1 , u2 , u3 } is an orthogonal set. Math 2331, Linear Algebra 3 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Sets: Theorem Theorem (4) Suppose S = {u1 , u2 , . . . , up } is an orthogonal set of nonzero vectors in Rn and W =span{u1 , u2 , . . . , up }. Then S is a linearly independent set and is therefore a basis for W . Partial Proof: Suppose c1 u1 + c2 u2 + · · · + cp up = 0 (c1 u1 + c2 u2 + · · · + cp up ) · = 0· (c1 u1 ) · u1 + (c2 u2 ) · u1 + · · · + (cp up ) · 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. 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 = {u1 , u2 , . . . , up } is an orthogonal basis for a subspace W of Rn and suppose y is in W . Find c1 ,. . .,cp so that y =c1 u1 + c2 u2 + · · · + cp up . Solution: = (c1 u1 + c2 u2 + · · · + cp up ) · y· y · u1 = (c1 u1 + c2 u2 + · · · + cp up ) · u1 y · u1 =c1 (u1 · u1 ) + c2 (u2 · u1 ) + · · · + cp (up · u1 ) y · u1 =c1 (u1 · u1 ) Similarly, c2 = ,. . . , =⇒ c1 = y·u1 u1 ·u1 cp = Math 2331, Linear Algebra 5 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Basis: Theorem Theorem (5) Let {u1 , u2 , . . . , up } 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 =c1 u1 + c2 u2 + · · · + cp up then cj = y · uj uj · uj (j = 1, . . . , p) Math 2331, Linear Algebra 6 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Basis: Example Example   3 Express y =  7  as a linear combination of the orthogonal basis 4       1 1 0    −1  ,  1  ,  0  .   0 0 1 Solution: y · u2 = u2 · u2 y · u1 = u1 · u1 y · u3 = u3 · u3 Hence y= u1 + u2 + Math 2331, Linear Algebra u3 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 b y= u·u u y·u z = y− u·u u =⇒ y · u − α (u · u) =0 =⇒ α= (orthogonal projection of y onto u) (component of y orthogonal to u) 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 u= u·u y to y Distance from y to the line through 0 and u = distance from b = kb y − yk = b y= Math 2331, Linear Algebra 9 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthonormal Sets Orthonormal Sets A set of vectors {u1 , u2 , . . . , up } in Rn is called an orthonormal set if it is an orthogonal set of unit vectors. Orthonormal Basis If W =span{u1 , u2 , . . . , up }, then {u1 , u2 , . . . , up } is an orthonormal basis for W . Recall that v is a unit vector if kvk = √ v·v = Math 2331, Linear Algebra √ vT v = 1. 10 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthonormal Matrix: Example Example Suppose U = [u1 u2 u3 ] where {u1 , u2 , u3 }   T  u1   [u1 u2 u3 ] =  U T U =  uT 2  uT 3 is an orthonormal set.      =      Orthogonal Matrix It can be shown that UU T = I . So U −1 = U T (such a matrix is called an orthogonal matrix). 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) = Math 2331, Linear Algebra 12 / 12