[SOLVED] Understanding Spanning

Mar 2010
175
3
Hey guys.

I have a simple question regarding spanning.

Obviously, if V = {(\(\displaystyle e_1, e_2, e_3, e_4\)) | (\(\displaystyle e_1, e_2, e_3, e_4\)) \(\displaystyle \in R^4\)}
V spans \(\displaystyle R^4\).

Can I say that V spans \(\displaystyle R^3\)??
 
Oct 2009
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Hey guys.

I have a simple question regarding spanning.

Obviously, if V = {(\(\displaystyle e_1, e_2, e_3, e_4\)) | (\(\displaystyle e_1, e_2, e_3, e_4\)) \(\displaystyle \in R^4\)}
V spans \(\displaystyle R^4\).

Can I say that V spans \(\displaystyle R^3\)??

Of course not: \(\displaystyle \mathbb{R}^3\) is not even a subset of \(\displaystyle \mathbb{R}^4\) and thus no element of V is contained in it!

Tonio
 
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Mar 2010
175
3
Of course not: \(\displaystyle \mathbb{R}^3\) is not even a subset of \(\displaystyle \mathbb{R}^4\) and thus no element of V is contained in it!
So basically,

(1,0,0,0) \(\displaystyle \in R^4\)

but

(1,0,0) is never \(\displaystyle \in R^4\).

Is it possible do draw a vector with 2 components in a three dimensional space?

and vice versa,

can you draw (1,0,0) in a two-dimensional plane???
 

dwsmith

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Mar 2010
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So basically,

(1,0,0,0) \(\displaystyle \in R^4\)

but

(1,0,0) is never \(\displaystyle \in R^4\).

Is it possible do draw a vector with 2 components in a three dimensional space?

and vice versa,

can you draw (1,0,0) in a two-dimensional plane???
In \(\displaystyle \mathbb{R}^2\), vectors are of the form \(\displaystyle (x,y)\); therefore, how can a vector of the form \(\displaystyle (x,y,z)\in\mathbb{R}^2\)?
 
Last edited:
Mar 2010
175
3
In , vectors are of the form ; therefore, how can a vector of the form ?
I dunno. But I would assume a two-dimensional plane (1,3) would be found in the (x,y) plane of (x,y,z) space...

In everyday life, we live in three-dimensional space containing two-dimensional things...Unless you count time as a 4th dimension...But let's not digress...
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
I dunno. But I would assume a two-dimensional plane (1,3) would be found in the (x,y) plane of (x,y,z) space...

In everyday life, we live in three-dimensional space containing two-dimensional things...Unless you count time as a 4th dimension...But let's not digress...
(1,3) Is not the same as (1,3,0)