Show that 𝑊 = {(𝑥, 4𝑥, 3𝑥) ∈ ℝ2 |𝑥 ∈ ℝ} is a subspace of ℝ 3 . Also find a basis for subspace 𝑈 of ℝ 3 which satisfies 𝑊 ⊕ 𝑈 = ℝ3
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Expert's answer
2022-05-10T11:49:32-0400
In order to prove that W is a subspace of R3 we will use a subspace criteria. Namely, it is enough to prove that u+w∈W for any two vectors u,w∈W and αu∈W for any α∈R. Suppose that u,w∈W. It means that u=(x,4x,3x) and w=(y,4y,3y) where x,y∈R. We obtain: u+w=(x+y,4(x+y),3(x+y)). Denote z:=x+y. We receive: (z,4z,3z)∈W. Multiply a vector u by α: αu=(αx,4αx,3αx). Denote αx=a. We receive: αu=(a,4a,3a)∈W. Thus, the criteria holds. Remind that W⊕U={W+U∣W∩U=∅}. I.e., the direct sum consists of sums of vectors from W and U. Consider a vector (1,4,3). it belongs to W. The matrix ⎝⎛100410301⎠⎞ is non-degenerate. The determinant is 1. It means that (1,4,3), (0,1,0),(0,0,1) is a basis in R3. Set U={(0,x,y)∣x,y∈R}. W∩U=∅, since (1,4,3),(0,1,0),(0,0,1) are linearly independent. W⊕U=R3, since (1,4,3),(0,1,0)(0,0,1) is a basis.
Answer: U={(0,x,y)∣x,y∈R}. The basis of U is: (0,1,0),(0,0,1).
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