Does S Span R^3? Determining Span and Linear Independence

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In summary, the span of S is the set of all possible linear combinations of the vectors in S, which can be used to generate a space by multiplying each vector by a scalar and adding them together. To determine if the span of S spans R^3, one must check if every vector in R^3 can be written as a linear combination of the vectors in S. A set of vectors can span more than one space because the span is defined as the set of all possible linear combinations. If the span of S does not span R^3, it means that there are some vectors in R^3 that cannot be written as a linear combination of the vectors in S. However, the span of S can also be equal to R
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physics=world
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1. Determine whether the set a) spans r3 and b) is linearly independent.
S = {(1,0,0) , (0,1,0), (0,0,1), (2,-1,0)}



Homework Equations





3. For Span, I put it in a matrix:

1 0 0 2
0 1 0 -1
0 0 1 0

From there I concluded that the dimension of the column equal to the rank of the matrix, which is 3. So, it spans R3.

Is this right?
 
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hi physics=world! :smile:
physics=world said:
… I concluded that the dimension of the column equal to the rank of the matrix, which is 3.

sorry, i don't understand :redface: …

can you please state this more clearly?​

(and anyway, isn't the answer obvious from looking at just the first three vectors?)
 

1. What is the definition of "span of S"?

The span of S refers to the set of all possible linear combinations of the vectors in S. In other words, it is the space that can be generated by multiplying each vector in S by a scalar and adding them together.

2. How can I determine if the span of S spans R^3?

To determine if the span of S spans R^3, you can check if every vector in R^3 can be written as a linear combination of the vectors in S. This means that for every point (x,y,z) in R^3, there must exist scalars a, b, and c such that a*v1 + b*v2 + c*v3 = (x,y,z), where v1, v2, and v3 are the vectors in S.

3. Can a set of vectors span more than one space?

Yes, a set of vectors can span more than one space. This is because the span of a set of vectors is defined as the set of all possible linear combinations of those vectors, so it can potentially span multiple spaces depending on the vectors in the set.

4. What does it mean if the span of S does not span R^3?

If the span of S does not span R^3, it means that there are some vectors in R^3 that cannot be written as a linear combination of the vectors in S. This could indicate that the set of vectors in S is not large enough to span the entire space of R^3.

5. Can the span of S be equal to R^3?

Yes, the span of S can be equal to R^3. This would mean that every vector in R^3 can be written as a linear combination of the vectors in S, thus making S a basis for R^3. However, this is not always the case and the span of S may only be a subset of R^3.

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