Define a subspace of a vector space V . Take the set of vectors in Rn...

Define a subspace of a vector space V . Take the set of vectors in Rn such that th
coordinates add up to 0. I that a subspace. What about the set whose coordinates add
up to 1. Explain your answers.

Solutions

Expert Solution

1.Let V be a vector space and let U be a subset of the set V. If U is also a vector space, then U is said to be a subspace of the vector space V.

2. Let X = (a₁,a₂,…,a_n) and Y = (b₁,b₂,…,b_n) be 2 arbitrary vectors in W, a subset of the set Rⁿ such that the coordinates of both X and Y add up to 0. Also, let k be an arbitrary real scalar. Then X+Y = (a₁,a₂,…,a_n) + (b₁,b₂,…,b_n) = (a₁+b₁,a₂+b₂,…,a_n+b_n). Also, (a₁+b₁)+(a₂+b₂)+…+(a_n+b_n) = (a₁+a₂+…+a_n)+(b₁+b₂+…+b_n)= 0+0 = 0. Hence W is closed under vector addition. Further, kX=k(a₁,a₂,…,a_n)= (ka₁,ka₂,…,ka_n). Also, ka₁+ka₂+ …+ ka_n = k(a₁+a₂+…+a_n) = k.0 = 0. Hence W is closed under scalar multiplication. Further, the zero vector 0 is apparently in W as its coordinates add up to 0. Therefore, W is a vector space and hence W is a subspace Rⁿ.

3. . Let X = (a₁,a₂,…,a_n) and Y = (b₁,b₂,…,b_n) be 2 arbitrary vectors in W, a subset of the set Rⁿ such that the coordinates of both X and Y add up to 1. Then X+Y = (a₁,a₂,…,a_n) + (b₁,b₂,…,b_n) = (a₁+b₁,a₂+b₂,…,a_n+b_n).Further, , (a₁+b₁)+(a₂+b₂)+…+(a_n+b_n) = (a₁+a₂+…+a_n)+(b₁+b₂+…+b_n)=1+1 = 2. Hence W is not closed under vector addition so that W is not a vector space and hence W is not a subspace Rⁿ.

milcah answered 1 year ago

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