Linear Algebra: Understanding Spans and Proving Inclusion in Vector Spaces

In summary: Therefore, in summary, the proof shows that the span of S is a subset of the span of T, which follows from the definitions of span(S) and span(T).
  • #1
RogerDodgr
20
0

Homework Statement


http://www.sudokupuzzles.net/IMG_0032.gif [Broken]


Homework Equations


I think I get the basic concept of spans (all possible combinations of vectors with all possible scalers).


The Attempt at a Solution

:blushing:
It seems obvious that the span of S would have to be in the span of T, I don't understand what is left to "prove". I have not done a lot of proofs. I don't know where to begin with this problem.
 
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  • #2
To prove a set A is a subset of some other set B, you must show that every element of A is also an element of B.

In your case, this will follow very easily from the definitions of span(S) and span(T).
 
  • #3
http://www.sudokupuzzles.net/IMG_0035.jpg [Broken]
Thank you quasar987,
I don't know how I could better state what seems obvious.

Maybe it should say 'All possible linear combinations...'

I won't be including the definition (or my misspelling, the definition was copied from my textbook). Sorry if this seems dumb; maybe I'm not clear what proofs are about; the premise of the question almost seems like proof.
 
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  • #4
I understand how you feel perfectly. Feels like I was there yesterday!

But stick to it... as you read more proofs and attempt to write some yourself, you will eventually see a pattern in the techniques used and you will pick up the proper vocabulary for writing proofs.

In the meantime, I give you this to feed on. Compare my proof to yours.

Proof: Let [tex]s=c_1v_1+...+c_kv_k[/tex] be an element of span(S).

We want to show that s is an element of span(T) also.

Recall that span(T) is the set of all elements of the form [tex]t=d_1v_1+...+d_mv_m[/tex]. In particular, for [tex]d_1=c_1,...,d_k=c_k[/tex] and [tex]d_{k+1}=...=d_m=0[/tex], we get that [tex]c_1v_1+...+c_kv_k+0v_{k+1}+...+0v_m=c_1v_1+...+c_kv_k=s[/tex] is an element of span(T).

Since the element s is arbitrary, it follows that all elements of span(S) are in span(T); that is to say, span(S) is a subset of span(T).
 
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1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with linear equations, vector spaces, and linear transformations. It is used to solve systems of linear equations and study the properties of vector spaces and their transformations.

2. What is a span in linear algebra?

A span in linear algebra is the set of all possible linear combinations of a given set of vectors. It represents all the possible directions and magnitudes that can be obtained by adding different combinations of the original vectors.

3. How do you determine if a vector is in a span?

To determine if a vector is in a span, you can set up a system of equations using the given vectors and the target vector. If the system is consistent (has a solution), then the target vector is in the span. If the system is inconsistent (has no solution), then the target vector is not in the span.

4. What is the difference between a span and a subspace?

A span is a set of vectors that can be obtained by linearly combining a given set of vectors. A subspace, on the other hand, is a set of vectors that satisfies three conditions: closure under addition, closure under scalar multiplication, and contains the zero vector. In other words, a subspace is a subset of a vector space that is also a vector space itself.

5. How is linear algebra used in real life?

Linear algebra has many real-life applications, including engineering, physics, economics, and computer graphics. It is used to solve systems of equations, analyze data, and model real-world scenarios. For example, linear algebra is used to design computer algorithms, optimize resource allocation, and create computer-generated images.

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