0
$\begingroup$

Suppose $a$ lies in the span of a set of independent vectors $E$. Now, if $a=b+c$, is it also the case that $b$ and $c$ lie in the span o the same set of vectors $E$?

if the question is obscure, please let me know. thanks in advance.

  • 1
    The question is not clear. "$E$ is the basis for $a$" only makes sense if $a$ is$a$vector space, and even then it should be "$E$ is a basis for $a$". But from the other things you write, it seems that $a,b,c$ are vectors, not vector spaces. You have to figure out what you actually mean here.2012-05-22

1 Answers 1

2

No. For example, try $b = e_1 - e_2$ and $c = e_1 + e_2$ where $e_1$ and $e_2$ are independent vectors. Then $a = 2 e_1$ is a linear combination of $e_1$, but $b$ and $c$ are not.

  • 0
    thanks sir. Really its very simple.2012-05-22