Homework SourseLet W be the subspace spanned by 1 1 1 1 3 3 5 5 3 5 1 3 Not
Let W be the subspace spanned by {[1 -1 1 1], [3 -3 5 5], [3 -5 1 3]}. Note that this basis is not orthogonal. Find the orthogonal projection of [-4 8 0 0] into W.![Let W be the subspace spanned by {[1 -1 1 1], [3 -3 5 5], [3 -5 1 3]}. Note that this basis is not orthogonal. Find the orthogonal projection of [-4 8 0 0] int](/WebImages/41/let-w-be-the-subspace-spanned-by-1-1-1-1-3-3-5-5-3-5-1-3-not-1127580-1761601611-0.webp "Let W be the subspace spanned by {[1 -1 1 1], [3 -3 5 5], [3 -5 1 3]}. Note that this basis is not orthogonal. Find the orthogonal projection of [-4 8 0 0] int")
Solution
Let u1 = (1,-1,1,1)T , u2 = (3,-3,5,5)T, u3 = (3, -5,1, 3)T and v = ( -4,8,0,0)T. Then Proju1(v)=[(v.u1)/(u1.u1)]u1 = [(-4-8)/(1+1+1+1)]u1= (-3)(1,-1,1,1)T = (-3,3, -3,-3)T.
Proju2(v)=[(v.u2)/(u2.u2)]u2 = [( -12-24)/(9 +9+25+25)]u2 =( -36/68)u2 = (9/17)(3,-3,5,5)T = (-27/17,27/17, -45/17,-45/17)T.
Also, Proju3 (v)=[(v.u3)/(u3.u3)]u3=[(-12-40)/(9+25+1+9)]u3=(-52/44)u3=(-13/11)(3,-5,1,3)T = ( -39/11, 65/11,-13/11, -39/11)T
Then, the orthogonal projection of v onto W is (-3,3, -3,-3)T +(-27/17,27/17, -45/17,-45/17)T+ (-39/11, 65/11,-13/11, -39/11)T = ( -1521/187, 1963/187, -1277/187, -1719/187)T
