I've been asked to show that the dimension of a certain subspace (intersection of 2 subspaces) is actually 1. My understanding is that the dimension is the number of vectors in the basis of a subspace. Given that $ X = \langle v \rangle$ and $v$ is given explicitly, I don't know what more to do to answer the question. How do I show something that seems to require little more than observation?
Or perhaps, I simply do not understand.
Let me show you the question:
I discovered that given two subspaces $U_1, U_2$ the following was true:
$$ U_1 \cap U_2 = \langle (1, 2, 0, 3) \rangle$$ (a column vector)
And the follow up question says show that $ \dim (U_1 \cap U_2) = 1$ and $ \dim (U_2) = 2$ where $U_2 = \langle (1, 2, 0, 3), (1, 0, 1, 1) \rangle$ (also column vectors). Now to me its pretty obvious that $ \dim (U_1 \cap U_2) = 1$ and $ \dim (U_2) = 2$ are true just by looking at the findings.
Or should I first go about showing that these sets are indeed the basis of their respective subspaces and then state that these are one-/two- element sets and consequently the statements about the dimensions are true or...?