Problem:
Determine which of the following are subspaces of ${\bf R}^3$.
All vectors of the form $(a, b, c)$, where $b = a + c + 1$.
My answers:
Thought process: to show if a set of vectors called W is a subspace, it must follow Axiom 1 and 6, closed under addition and scalar multiplication respectively.
$(a, b, c)$, where $b = a + c + 1$.
$(a,a+ c + 1, c) + (a,a + c + 1, c) = (2a,2a + 2c + 2, 2c) $
No. It is not closed under addition because the form completely changed specifically the y component of the vector completely changed.
$2(a, a + c + 1,c) = (2a, 2a + 2c + 2, 2c)$.
No. It is not closed under scalar multiplication either because the form completely changed specifically the y component of the vector completely changed.
Since it is not closed under addition and scalar multiplication, I can say it is not a subspace of ${\bf R}^3$.
Is this the right way to think about figuring out if a set of vectors is a subspace by thinking of the final result has the same form as the original vector?
Best Answer
Generally, the best way to show that a set isn't a subspace is with a specific example of one of the necessary properties failing. In this case, it is easy to see that $(0,0,0)$ isn't in the set, and so it isn't a subspace. What you've done is okay though.