Take any vector A in (R^3. Let W be the set of all vectors B in (R^3 where B.A = 0. Discuss whether W is a subspace of (R^3 or not.

Answer 1

To determine whether W is a subspace of R^3, we need to ensure that it satisfies three conditions:

1) W contains the zero vector
2) W is closed under vector addition
3) W is closed under scalar multiplication.

1) W contains the zero vector:
Let A = (a1, a2, a3). To satisfy B.A = 0, we can choose B = (0, 0, 0), which is the zero vector. Therefore, W contains the zero vector.

2) W is closed under vector addition:
Consider two vectors B1 and B2 in W, such that B1.A = B2.A = 0.
Now we need to show that the sum of B1 and B2, denoted as B1 + B2, also satisfies B1 + B2 . A = 0.

(B1 + B2) . A = B1 . A + B2 . A (by the distributive property)
= 0 + 0 (since B1.A = B2.A = 0)
= 0

Therefore, B1 + B2 also satisfies B1 + B2 . A = 0, and W is closed under vector addition.

3) W is closed under scalar multiplication:
Consider a vector B in W and a scalar c. We need to show that cB also satisfies (cB).A = 0.

(cB).A = c(B.A) (by the distributive property)
= c(0) (since B.A = 0)
= 0

Therefore, cB also satisfies (cB).A = 0, and W is closed under scalar multiplication.

Since W satisfies all three conditions, we can conclude that W is a subspace of R^3.