Prove that there is a subset of $T$, which, together with $S$, is again a basis of $E$. 
Suppose $S$ is a set of linearly independent vectors in $E$, and suppose $T$ is a basis of $E$. Prove that there is a subset of $T$, which, together with $S$, is again a basis of $E$. --Linear Algebra by Werner Greub.

Where should I start with please?
 A: I'll assume your space $E$ is finite dimensional.
I'll do it by induction. Let $S=\{v_1,\dots,v_m\}$ and, for $0\le k\le m$, set 
$$
S_k=\{v_j:1\le j\le k\}.
$$
In particular, $S_0=\emptyset$.
We'll prove that, if $k\le n$, where $n=\dim E$, there is a subset $T_k$ of $T$ such that $S_k\cup T_k$ is a basis of $E$.
The base step consists in choosing $T_0=T$. Now suppose we have the thesis for $k$; if $k=n$, we are done, so we can suppose $k<n$.
By the induction hypothesis,
$$
v_{k+1}=\alpha_1 v_1+\dots+\alpha_k v_k+\beta_1 w_1+\dots+\beta_{n-k}w_k
$$
where $T_k=\{w_1,\dots,w_{n-k}\}$. One of the coefficients $\beta_1,\dots,\beta_{n-k}$ must be nonzero, since $S$ is linearly independent. We can rename the indices so that $\beta_1\ne0$, which allows us to write the above relation as
$$
w_1=\alpha_1'+\dots+\alpha_k'+\alpha_{k+1}'v_{k+1}+\beta_2'w_2+\dots+\beta_{n-k}'w_{n-k}
$$
It's easy to prove now that $T_{k+1}=\{w_2,\dots,w_{n-k}\}$ is a good choice for completing our task.
A: This is problem 9 of section §3 of Chapter 1 in the last (4th) edition of Greub's "Linear Algebra".
Here's my answer: 
(Note that the case of finite $S$ amounts to Steinitz's replacement theorem). 
Let $Z$ be the subspace of $E$ generated by $S$. If $Z=E$, then we may take $\emptyset\subseteq T$ and we are done. So assume $Z\subsetneq E$. It follows that some element of $T$ lies outside $Z$ (otherwise any vector in $E$, being a linear combination of the elements of $T$, would also lie in $Z$, contradicting $Z\subsetneq E$), say $v\in T\smallsetminus Z$. 
Let $\mathscr{M}(T,Z) \doteq\left\{ T'\subseteq T\mid T'\text{ is linearly independent mod }Z\right\}$. 
It is clear that $\left\{ v\right\} \in\mathscr{M}(T,Z)$ (since $v\notin Z$, we have $\lambda v\in Z\iff\lambda=0$), so $\mathscr{M}(T,Z)$ is nonempty. Now $\mathscr{M}(T,Z)$ is partially ordered by inclusion, and every chain $\mathscr{C}_{i}\subseteq\mathscr{M}(T,Z)$ has an upper bound $\bigcup_{T'\in\mathscr{C}_{i}}T'$. Indeed, every finite subset of $\bigcup_{T'\in\mathscr{C}_{i}}T'$ is contained in some element of the chain $\mathscr{C}_{i}$ and hence linearly independent $\text{mod }Z$, so $\bigcup_{T'\in\mathscr{C}_{i}}T'$ is itself linearly independent $\text{mod }Z$, whence $\bigcup_{T'\in\mathscr{C}_{i}}T'\in\mathscr{M}(T,Z)$. It follows by Zorn's lemma that there's a maximal element (with respect to inclusion) of $\bigcup_{T'\in\mathscr{C}_{i}}T'\in\mathscr{M}(T,Z)$, call it $T'_{\Omega}$. Now take any $x\in E\smallsetminus T'_{\Omega}$. Then, by maximality of $T'_{\Omega}$, the set $T'_{\Omega}\cup\left\{ x\right\}$ must be linearly dependent $\text{mod }Z$, so for some scalars $\xi^{v}$, not all zero, we have: 
$\xi x + \sum_{x_{v}\in T'_{\Omega}}\xi^{v}x_{v}\in Z$. 
But if we had $\xi=0$, it would follow that, for some scalars $\xi^{v}$, not all zero, we have $\sum_{x_{v}\in T'_{\Omega}}\xi^{v}x_{v}\in Z$, contradicting the linear independence $(\text{mod }Z)$ of $T'_{\Omega}$. Hence $\xi\neq0$. It now follows that
$\xi x-\sum_{x_{v}\in T'_{\Omega}}(-\xi^{v})x_{v}\in Z$
$x-\sum_{x_{v}\in T'_{\Omega}}\xi^{-1}(-\xi^{v})x_{v}\in Z$.
Let $z\in Z$ be the left side in the last display. Then, for any $x\in E\smallsetminus T'_{\Omega}$, and setting $\eta^{v}=(\xi^{x})^{-1}(-\xi^{v})$, we have: 
$x=\sum_{x_{v}\in T'_{\Omega}}\eta^{v}x_{v}+z$
$x=\sum_{x_{v}\in T'_{\Omega}}\eta^{v}x_{v}+\sum_{y_{w}\in S}\alpha^{w}y_{w}$
for some scalars $\alpha^{w}\in\varGamma$. Thus $T'_{\Omega}\cup S$ generates all $x\in E\smallsetminus T'_{\Omega}$. But, trivially, it generates all of $T'_{\Omega}$ as well. So $T'_{\Omega}\cup S$ is a system of generators for $E$. Now suppose that 
$\sum_{x_{v}\in T'_{\Omega}}\beta^{v}x_{v}+\sum_{y_{w}\in S}\gamma^{w}y_{w}=0$
Then 
$\sum_{x_{v}\in T'_{\Omega}}\beta^{v}x_{v}=-\sum_{y_{w}\in S}\gamma^{w}y_{w}\in Z$ 
so, since $T'_{\Omega} \in \mathscr{M}(T,Z)$ is linearly indepedent $\text{mod } 
Z$, we have
$\sum_{x_{v}\in T'_{\Omega}}\beta^{v}x_{v}=0$. Since $T'_{\Omega}\subseteq T$, it is a linearly independent set, so all $\beta^{v}$'s are zero. Since $S$ is also linearly independent, this in turn implies that all $\gamma^{w}$'s are zero as well. So the system of generators $T'_{\Omega}\cup S$ is linearly independent, and hence a basis. This completes the proof.
