Question3. Determine whether the sets defined by the following vectors are subspaces of $\mathbf{R}^{3}$. (a) $(a, b, 2 a+3 b)$

Studdy Solution
Check if the set is closed under scalar multiplication.
Take an arbitrary vector $(a, b, 2a + 3b)$ and a scalar $c$.
Multiply the vector by the scalar:
$$c \cdot (a, b, 2a + 3b) = (ca, cb, c(2a + 3b))$$
$$= (ca, cb, 2(ca) + 3(cb))$$
The result is in the same form as the original vector, so the set is closed under scalar multiplication.
Since the set satisfies all three conditions, it is a subspace of $\mathbf{R}^3$.