Question

Linear Algebra

Write x as the sum of two vectors, one is Span {u_{1},
u_{2}, u_{3}} and one in Span {u4}. Assume that
{u_{1},...,u_{4}} is an orthogonal basis for
R^{4}

u_{1} = [0, 1, -6, -1] , u_{2} = [5, 7, 1, 1],
u_{3} = [1, 0, 1, -6], u_{4} = [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.u_{1})/(u_{1}.u_{1})]u_{1} =
[(0-9-24+0)/(0+1+36+1)]u_{1} = -(33/38) (0, 1, -6, -1) =
(0,-33/38, 99/19, 33/38); proj_{u2}(x) =
[(x.u_{2})/(u_{2}.u_{2})]u_{2} =
[(70-63+4+0)/(25+49+1+1)]u_{1} = (11/76) (5, 7, 1, 1) =
(55/76,77/76,11/76,11/76); and proj_{u3}(x) =
[(x.u_{3})/(u_{3}.u_{3})]u_{3} =
[(14+0+4+0)/(1+0+1+36)]u_{3}= (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
{u _{1}, u_{2}, u_{3}} and u =
(76/139)u_{4} is in span {u_{4}}.**

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weights. If not, explain why.
v =
2
4
2
, u1 =
1
1
0
, u2 =
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1
-1
, u3 =
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2
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