Proof of Equality for X1, X2, X3 Vector-Space Spans

  • Thread starter EvLer
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In summary: If X_1 and X_2 are dependent then span(X_1+X_2) must equal span(X_1,X_2). Since X_1 and X_2 are not independent, X_1+X_2 does not equal X_1+X_2.
  • #1
EvLer
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Hi,
I have a T/F which I need to prove.

X1, X2, X3 belong to vector-space V.
Y1 = X1 + X2, Y2 = X3.
Span{Y1, Y2} is contained in but not equal to span{X1, X2, X3}.

I am not sure which one it is:
since y-span can be represented as span{X1 + X2, X3} it may be false, but then if all spans are subspaces, these two subspaces are not of the same dimension, i.e. they are not equal, then the statement is true. Obviously one of my reasonings is wrong. :rolleyes:
Could someone clear up this for me?
Thank you in advance.
 
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  • #2
EvLer said:
Hi,
I have a T/F which I need to prove.

X1, X2, X3 belong to vector-space V.
Y1 = X1 + X2, Y2 = X3.
Span{Y1, Y2} is contained in but not equal to span{X1, X2, X3}.

I am not sure which one it is:
since y-span can be represented as span{X1 + X2, X3} it may be false, but then if all spans are subspaces, these two subspaces are not of the same dimension, i.e. they are not equal, then the statement is true. Obviously one of my reasonings is wrong. :rolleyes:
Could someone clear up this for me?
Thank you in advance.


It's false. Let X_i=0. The span of both is {0}.
 
  • #3
phoenixthoth said:
It's false. Let X_i=0. The span of both is {0}.
But what if X_i is not lin. dep? In which case, {0} would not work...
 
  • #4
If X_1 and X_2 are linearly dependent then clearly span(X_1+X_2)=span(X_1,X_2),
so the assertion is false in general.

If X_1 and X_2 are lin. indep., X_3 could still be dep. on X_1 + X_2.
 
  • #5
Galileo said:
If X_1 and X_2 are linearly dependent then clearly span(X_1+X_2)=span(X_1,X_2),
Actually, that is what I was not sure about, because this:
span{X1, X2} = span{X1, X2, c1X1 + c2X2} I see.
But this:
span{X1, X2} = span{X1 + X2}
I don't quite. Could you outline the proof?
If I look at it from the stand-point of dimension, first one has dim. of 2, second -- 1, which I take to mean that they are not equal, even if one is lin. comb. of the other.
Am I totally off?
Thank you very much.
 
  • #6
Of X_1 and X_2 are dependent, then they span a line, one is simply a scalar multiple of the other.

Use the definition of linear dependence to prove this:
X_1 and X_2 are linearly dependent means X_1=cX_2 for some c.
 

FAQ: Proof of Equality for X1, X2, X3 Vector-Space Spans

What is "Proof of Equality for X1, X2, X3 Vector-Space Spans"?

"Proof of Equality for X1, X2, X3 Vector-Space Spans" is a mathematical concept that refers to showing that two sets of vectors, X1, X2, X3 and Y1, Y2, Y3, span the same vector space. In other words, any vector in the vector space can be expressed as a linear combination of the vectors in both sets.

Why is it important to prove equality for vector-space spans?

Proving equality for vector-space spans is important because it helps us understand the relationships between different sets of vectors and their span. It also allows us to identify when two seemingly different sets of vectors actually span the same vector space, which can greatly simplify mathematical calculations and proofs.

What are the key steps in proving equality for vector-space spans?

The key steps in proving equality for vector-space spans are: 1) Showing that the two sets of vectors, X1, X2, X3 and Y1, Y2, Y3, span the same vector space. This can be done by showing that any vector in the vector space can be expressed as a linear combination of the vectors in both sets. 2) Showing that the two sets are linearly independent, meaning that no vector in either set can be expressed as a linear combination of the other vectors. 3) Showing that the two sets have the same number of vectors, or the same dimension.

What are some common techniques used to prove equality for vector-space spans?

Some common techniques used to prove equality for vector-space spans include: 1) Using the definition of span to show that any vector in the vector space can be expressed as a linear combination of the vectors in both sets. 2) Using the definition of linear independence to show that the two sets are linearly independent. 3) Using the fact that the two sets have the same number of vectors, or the same dimension, to prove equality.

Are there any real-world applications of proving equality for vector-space spans?

Yes, there are many real-world applications of proving equality for vector-space spans. For example, in engineering, it is important to understand the relationships between different sets of vectors and their span in order to design efficient structures and systems. In computer graphics, proving equality for vector-space spans can help create more realistic 3D images. In physics, this concept is used to understand the relationships between different physical quantities and their units of measurement.

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