# Thread: Subspace spanned by subspace

1. ## Subspace spanned by subspace

Hi,

problem:
Suppose that x and y are vectors and M is a subspace in a vector space V;
let K be the subspace spanned by M and x, and let K be the subspace spanned by M and y.
Prove that if y is in K but not in M, then x is in K.

attempt:
So, K is the set of all linear combinations of elements in M and the vector y.
K is also the set of all linear combination of elements in M and the vector x.
This definition of K makes me think that x is always in K.
What am I missing here?

Thanks!

2. Hi,
Originally Posted by Mollier
problem:
Suppose that x and y are vectors and M is a subspace in a vector space V;
let K be the subspace spanned by M and x, and let K be the subspace spanned by M and y.
Prove that if y is in K but not in M, then x is in K.
It does not make sense : $K$ is at the same time the subspace spanned by $M$ and $x$ and the subspace spanned by $M$ and $y$... but these subspaces may be different !

Here is what I think they meant :
problem:
Suppose that $x$ and $y$ are vectors and $M$ is a subspace in a vector space $V$;
let $K_x$ be the subspace spanned by $M$ and $x$, and let $K_y$ be the subspace spanned by $M$ and $y$.
Prove that if $y$ is in $K_x$ but not in $M$, then $x$ is in $K_y$.

3. Hi!

We then have:


\begin{aligned}
\mathbb{K}_x=&\;span\{\mathbb{M},x\}\\
\mathbb{K}_y=&\;span\{\mathbb{M},y\}
\end{aligned}

If $y\in\mathbb{K}_x$ but $y\notin\mathbb{M}$ then $x\in\mathbb{M}$ and therfore $x\in\mathbb{K}_y$.

Is this any better?

Thanks.

4. Gentle bump.