Let V=Rn be a vector space and W = {(a1, a2,…, an) such that a2=a12. Check whether W isa subspace of V or not, with respect to usual addition and scalar multiplication.
Question
Let V=Rn be a vector space and W = {(a1, a2,…, an) such that a2=a12. Check whether W isa subspace of V or not, with respect to usual addition and scalar multiplication.
Solution
To check if W is a subspace of V, we need to verify three properties:
- The zero vector of V is in W.
- W is closed under vector addition.
- W is closed under scalar multiplication.
Let's check each property:
-
The zero vector in R^n is the vector with all components equal to zero. In this case, a2 = a1^2 = 0^2 = 0, so the zero vector is in W.
-
Let's take two vectors in W, say (a1, a2,..., an) and (b1, b2,..., bn). We know that a2 = a1^2 and b2 = b1^2. If we add these two vectors, we get (a1+b1, a2+b2,..., an+bn). For this to be in W, we need (a2+b2) = (a1+b1)^2. But this is not necessarily true, because (a1+b1)^2 = a1^2 + 2a1b1 + b1^2, and this is not equal to a1^2 + b1^2 unless a1 = b1 = 0. So, W is not closed under vector addition.
Since W is not closed under vector addition, we don't need to check the third property. We can conclude that W is not a subspace of V.
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