Question

# Linear Algebra Write x as the sum of two vectors, one is Span {u1, u2, u3}...

Linear Algebra

Write x as the sum of two vectors, one is Span {u1, u2, u3} and one in Span {u4}. Assume that {u1,...,u4} is an orthogonal basis for R4

u1 = [0, 1, -6, -1] , u2 = [5, 7, 1, 1], u3 = [1, 0, 1, -6], u4 = [7, -5, -1, 1], x = [14, -9, 4, 0]

x =

(Type an integer or simplified fraction for each matrix element.)

We have proju1(x) = [(x.u1)/(u1.u1)]u1 = [(0-9-24+0)/(0+1+36+1)]u1 = -(33/38) (0, 1, -6, -1) = (0,-33/38, 99/19, 33/38); proju2(x) = [(x.u2)/(u2.u2)]u2 = [(70-63+4+0)/(25+49+1+1)]u1 = (11/76) (5, 7, 1, 1) = (55/76,77/76,11/76,11/76); and proju3(x) = [(x.u3)/(u3.u3)]u3 = [(14+0+4+0)/(1+0+1+36)]u3= (9/19) (1, 0, 1, -6) = (9/19,0,9/19, -54/19).

Further, projW (x) = proju1(x) + proju2(x) + proju3(x) =(0,-33/38, 99/19, 33/38)+ (55/76,77/76,11/76,11/76)+ (9/19,0,9/19, -54/19)= (91/76,11/76,443/76,-139/76) = v(say)

Also, x-v =u (say) = (14, -9, 4, 0)- (91/76,11/76,443/76,-139/76) = (973/76,-695/76,-139/76,139/76).

Then x = v+u = (91/76,11/76,443/76,-139/76)+ (973/76,-695/76,-139/76,139/76), where v is in W = Span {u1, u2, u3} and u = (76/139)u4 is in span {u4}.

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