What does it mean for a span to be a Subspace of another span . For example, I have this question:  If $u, v$ belong to a vector space $V$ then span{$u, v$} = span{$u, u + v$}. True or False
So what we need to show is that span{$u, v$} is a subspace of span{$u, u + v$} and vice versa. And if it is the case, then they are equal. However I don't even understand what it would mean for a span to be a subspace of another span. Also, In the answer key they took an arbitrary vector "$w$" and assumed that it was an element of span{$u, v$} then proceeded to show that its also an element of span{$u, u + v$}. Then concluded that span{$u, v$} is a subspace of span{$u, u + v$}. I also don't understand why they concluded that.
 A: The span of $\{u,v\}$ is the set of all linear combinations of $u$ and $v$, i.e. all vectors that can be written as $\lambda_1 u + \lambda_2 v$ for $\lambda_1,\lambda_2$ in the corresponding field (usually $\mathbb{R}$ or $\mathbb{C}$). Having two subspaces given as a span is very useful because it allows to easily check the relation between the two subspaces. Namely, if you have $X := \mathrm{span}\{u,v\}$ and $Y := \mathrm{span}\{a,b\}$ and you can show that $u,v \in Y$, then necessarily $X \subseteq Y$. The reason for this is quite obvious. If you have any $x \in X$, then you can write it as a linear combination of $u$ and $v$: $x = \lambda_1 u + \lambda_2 v$. But since $u,v \in Y$, $u$ and $v$ can be written as a linear combination of $a$ and $b$: $u = \mu_1 a + \mu_2 b$, $v = \eta_1 a + \eta_2 b$. Now plug those into the formula for $x$ and you have $x$ as a linear comination of $a$ and $b$, hence $x \in Y$. As $x \in X$ was arbitrary, we get $X \subseteq Y$.
A: What you need to show is the following:

*

*span{$u,v$} $\subset$ span{$u,u+v$}

*span{$u,u+v$} $\subset$ span{$u,v$}

Now in general, when you have two sets, let's say $A$ and $B$, for which holds $A \subset B$, that means that $\forall x \in A \Rightarrow x \in B$. The same logic can be applied to your example. That's why in the key the arbitrary vector $w$ is taken.
What you need to do from here is (I'm going to start only the first inclusion, I'm leaving the second one to you):
Let's assume that $w \in$ span{$u,v$}. That means that $w$ can be written as $\lambda u+\mu v$, for some scalars $\lambda, \mu$. The next step is to show that $w$ can be also written as $\tilde{\lambda}u+\tilde{\mu}(u+v)$. If you can show this, than the first inclusion is done.
