Question

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.)

     

Answer 1

We have proj_u1(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); proj_u2(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 proj_u3(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, proj_W (x) = proj_u1(x) + proj_u2(x) + proj_u3(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}.