Could anyone explain me why Gaussian elimination method ( type 3: Add to one row a scalar multiple of another ) does not change linear span of vector set?
For example I have linear span of this set of vectors in $R^{3}$ defined as:
$<(1,4,2),(2,5,1)>$
Why can we say that if I do G3 changes for example:
$$2 * (1,4,2) + (2,5,1) = (4,13,5) $$ $$3*(4,13,5) + (1,4,2) = (13,43,17)$$
then $$<(1,4,2),(2,5,1)> = <(4,13,5),(13,43,17)> $$
On
Suppose you do a type 3 operation, under which $\langle v,w \rangle$ is replaced by $\langle v+aw,w \rangle$.
For any coefficients $r,s$ it follows that $$r\,(v+aw)+s\,w = r\,v + (ra+s)\,w $$ which shows that $\langle v+aw,w \rangle \subset \langle v,w \rangle$.
Also, for any coefficients $t,u$ we have $$tv + uw = t \, (v+aw) + (u-ta) \,w $$ which shows that $\langle v,w \rangle \subset \langle v+aw,w \rangle$.
Putting these together we have $$\langle v+aw,w \rangle = \langle v,w \rangle $$
A similar argument, which differs only by symmetry of notation, shows that $$\langle v,w+bv \rangle = \langle v,w \rangle $$
The span is the set of all linear combinations of given vectors. You could recover your previous vector by doing the opposite operation to what you just did, and that would still be a linear combination. So the set of stuff you could make before is still present. So everything you made before, you can still make, and everything you can make now, you could have made before.