Subspace spanned by subspace

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February 4th 2010, 09:27 PM
Mollier

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!
February 5th 2010, 12:36 AM
flyingsquirrel

Hi,

Quote:

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 :

Quote:

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 $K_y$ .
February 5th 2010, 11:58 PM
Mollier

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.
February 9th 2010, 07:47 PM
Mollier

Gentle bump.