Answer to Question #195080 in Linear Algebra for cayyy

Question

Let U be the subspace of R⁵ denoted by: U = { (x₁; x₂; x₃; x₄; x₅) ∈ R⁵ : x₁ = 3x₂ and x₃ = 7x₄

1)   Find a subspace W of R⁵ such that R⁵ = U ⊕ W.

Answer 1

since U is given by two linear independent equations in 5-dimensional space, then this is a subspace of dimension 3, that is, if we find a basis of the subspace U and complement it to a basis of 5-dimensional space.
That subspace with complementary vectors and the subspace U is a direct sum.

basis u (u1(1,3,0,0,0),u2(0,0,1,7,0)u3(0,0,0,0,1))

w(w1(1,-3,0,0,0),w2(0,0,1,-7,0))

if in the n-dimensional space there are n vectors such that it is possible to express the entire vector of the basis, then this set is also a basis

(u1+w1)/2=e1(1,0,0,0,0)

(u1-w1)/6=e2(0,1,0,0,0)

(u2+w2)/2=e3(0,0,1,0,0)

(u2-w2)/14=e4(0,0,0,1,0)

u3=e5(0,0,0,0,1)

u1.u2.u3.w1.w2 -basis R5

using the initial statement, we need 3 linear equations to write down the subspace W

w|x1=-3x2,x3=-7x4,x5=0