# spanning set for R^2

• Sep 8th 2008, 09:39 PM
apple12
spanning set for R^2
First, can anyone explain to me the meaning of spanning set? I am a little confused with it meaning. :(

Then please show me how to do the below questions. Thank you very much.

Determine whether any of the following sets are spanning sets for R^2 , considered as column matrices
1) S = { esub1 = [1] , esub2 = [0]}
........................[0]............. [1]

2) S = { esub1 = [1] , esub2 = [0], fsub1 = [1]}
........................[0]............. [1].............[1]

3) S = { fsub1 = [1] , fsub2 = [2]}
.......................[1]............. [2]

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note: e and f are vectors.
• Sep 8th 2008, 11:13 PM
CaptainBlack
Quote:

Originally Posted by apple12
First, can anyone explain to me the meaning of spanning set? I am a little confused with it meaning. :(

A spanning set for a (sub-) space $\displaystyle W$ is a set of elements of $\displaystyle W$ which spans $\displaystyle W$. A set $\displaystyle S \subset W$ spans $\displaystyle W$ if for all $\displaystyle w \in W$ $\displaystyle w$ is a linear combination of elements of $\displaystyle S$.

Quote:

Then please show me how to do the below questions. Thank you very much.

Determine whether any of the following sets are spanning sets for R^2 , considered as column matrices
1) S = { esub1 = [1] , esub2 = [0]}
........................[0]............. [1]
This is the standard basis for $\displaystyle R^2$, so spans $\displaystyle R^2$.

Quote:

2) S = { esub1 = [1] , esub2 = [0], fsub1 = [1]}
........................[0]............. [1].............[1]
This contains the standard basis and so spans $\displaystyle R^2$

Quote:

3) S = { fsub1 = [1] , fsub2 = [2]}
.......................[1]............. [2]
$\displaystyle [1,2]'$ cannot be written as a linear combination of elements of $\displaystyle S$, so $\displaystyle S$ does not span $\displaystyle R^2$.

RonL