Linear Algebra: Proving or Disproving Span Equivalence

  • Thread starter *best&sweetest*
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In summary, the conversation discusses a linear algebra exam question about proving or disproving that if the span of two sets, S and P, are equal, then S must equal P. The speaker provides a counterexample using specific vectors and explains that the statement is not always true. They also mention that the question seemed easy once they thought of the counterexample. The other speaker confirms that the example is correct and thanks the first speaker for their help.
  • #1
*best&sweetest*
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I have just had linear algebra exam, and one of the questions was to prove or disprove (give a counterexample) that if span(S) = span(P) then S=P.

I gave this example:

S={v} where vector v=(1,2), and P={u} where u= (2,4).
Then, span(S) = span (P) (line through origin parallel to vector (1,2)),
but S does not equal P.

Am I right?
Thank you!
 
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  • #2
Yes, that's perfectly correct. (And it was kind of a silly question!)
 
  • #3
Thank you! I spent 15 minutes proving that, and then decided to find a counterexample. I was confused because it seemed so easy once I got the idea. Thanks again!
 

What does "span(S)=span(P)" mean?

This notation means that the set of all possible linear combinations of vectors in S is equal to the set of all possible linear combinations of vectors in P. In other words, the span of S and the span of P are the same.

What does it mean if span(S)=span(P)?

If span(S)=span(P), it means that the two sets of vectors contain the same elements and can be used to create the same set of vectors through linear combinations.

Why is span(S)=span(P) important?

This equality is important because it shows that S and P have the same "spanning power" or the ability to create the same set of vectors. This can be useful in various applications in linear algebra and other fields of science.

Does span(S)=span(P) always imply that S=P?

No, it does not always imply that S=P. While the equality of spans means that the two sets are capable of creating the same set of vectors, it does not necessarily mean that the two sets are identical in terms of their individual vectors. S and P can still have different elements even though their spans are equal.

How can we determine if span(S)=span(P)?

To determine if span(S)=span(P), we can check if every vector in S can be created through a linear combination of vectors in P, and vice versa. If this is true, then span(S)=span(P). Alternatively, we can also use other methods such as Gaussian elimination or computing the rank of the matrix formed by the vectors in S and P.

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