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OpenStudy (he66666):

Linear algebra: span Which of the following vectors span R2(1x2 matrix)? (a) [1 2], [-1 1] (b) [0 0], [1 1], [-2 -2] Do I make the matrices into row reduced form in order to solve these? Like for (a), would it be like [1 2; -1 1], so [1 0; 0 1]?

OpenStudy (anonymous):

I think both span it.

OpenStudy (anonymous):

Sure the second one is not lin ind but it still spans

OpenStudy (kinggeorge):

The second set does not span R2

OpenStudy (anonymous):

Why not?

OpenStudy (kinggeorge):

Notice that [0,0] and [-2, -2] are both scalar multiples of [1, 1]. This means that the second set of vectors only spans the set of vectors [a, a] where a is some number in R.

OpenStudy (he66666):

How do you know if the 1xn matrices span it? I am confused

OpenStudy (he66666):

If it was an augmented matrix, I would determine whether it spans or not by its consistency but I don't know how to deal with this question..

OpenStudy (anonymous):

Oh yeah I didnt catch that. But if [0 0] was [1,2] then it would span r^2 right?

OpenStudy (kinggeorge):

If [0, 0] was [1, 2] it would span. @he66666 I can determine that the first set spans R2 because I can see that R2 is a two-dimensional vector space, and [1, 2] and [-1, 1] are two linearly independent vectors. This means that they form a basis, and so they span R2.

OpenStudy (anonymous):

What if you have a set of three vectors that are lin independent and are two dimensional. Does that set also span R^2?

OpenStudy (kinggeorge):

If the vectors are contained in R^3, I believe that the lin. independent vectors span a vector space that's isomorphic to R^2.

OpenStudy (kinggeorge):

For example, take the vectors [1 1 1] and [1 1 2]. These are obviously linearly independent, but they are not contained in R^2, so they can't actually span R^2. They would instead span some subspace of R^3 that is isomorphic to R^2.

OpenStudy (he66666):

I didn't learn independent vectors yet. But we learned that we should investigate the consistency of the linear system to determine whether a specific vector v belongs to span S, and using the reduced row form to see whether it spans. Is there another way to solve it?

OpenStudy (kinggeorge):

For the first one, I think you would want to rref [1 -1;2 1]. If one of the two rows is 0 after you rref it, they should be linearly independent (aka not in the span of the other).

OpenStudy (kinggeorge):

OS just crashed twice for me x.x By showing that the other vectors are scalar multiples of [1 1], you are showing that they are in the same span.

OpenStudy (turingtest):

perhaps another way to see it is that the vector [0,0] has no span, so we need only consider the other two vectors

OpenStudy (he66666):

Thanks guys :)

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