Question

Decide whether the given vector is a member of Span(S) for each given set S. If so, express it as a linear combination from S:

Answer 1

\[\huge S=\left\{ \frac1 x , \frac 1 {x^2}, \frac1 {x^3}\right\}\subseteq F((0,\infty)); f(x)= {2{x^2}-7x-10\over {x^3}}\]

Answer 2

\[\huge f(x)={2{x^2}-7x-10\over {x^3}}\]

Answer 3

Basically, you have to ask yourself, can:\[\frac{2x^2-7x-10}{x^3}\]be written as a linear combination of 1/x, 1/x^2, and 1/x^3? in other words, can you find scalars a, b, and c, such that: \[\frac{2x^2-7x-10}{x^3}=a\left(\frac{1}{x}\right)+b\left(\frac{1}{x^2}\right)+c\left(\frac{1}{x^3}\right)\]

Answer 4

Okay, so that's the "quote on quote" Dependence Equation, but how would I write it a a matrix to use the rref?

Answer 5

You dont need to create a matrix at all for this problem.

Answer 6

infact, this is an infinite dimensional vector space, so creating a matrix is pretty much out of the question >.<

Answer 7

Well, mainly because that's how I learned how to do it, distribute the variables, in this case you used a, b, and c, and then write the constants in one row, the terms with powers of 1 in the next etc.

Answer 8

right, that is correct when dealing with "regular" vector spaces. however, we have an abstract vector space before us. This calls for different techniques.

Answer 9

what do you suggest?

Answer 10

Here is an example. It will show you how to do this problem. I claim that:\[\frac{-3x^2-27x+2}{x^3}\]is in the span of those three functions. We can see this because:\[\frac{-3x^2-27x+2}{x^3}=\frac{-3x^2}{x^3}+\frac{-27x}{x^3}+\frac{3}{x^3}\]\[\frac{-3}{x}+\frac{-27}{x^2}+\frac{3}{x^3}=-3\cdot\frac{1}{x}-27\cdot\frac{1}{x^2}+3\cdot\frac{1}{x^3}\]So the function is a linear combination of 1/x, 1/x^2 and 1/x^3.

Answer 11

uh, I get the denominator...but how did you get the numerator? I looks like you did partial fraction decomposition...

Answer 12

All i did was divide it out. its like saying:\[\frac{5+4}{9}=\frac{5}{9}+\frac{4}{9}\]

Answer 13

ah, in that case, 1) you're a fast typer to be able to type the fraction operator (using that annoying equation editor)
2) you have a typo.

Answer 14

oh oh oh, yes, that 3 should be a 2. my bad.

Answer 15

Ok, so since f(x) is a linear combination of the three "vectors" in S, it is in the span of the set?

Answer 16

that is correct. you could even answer the "regular" vector space questions in this manner. if i asked if (1,3,5) was in the span of {(1,1,1) ,(1,2,3)}, you could either create the matrix, rref, etc, or just note that (1,3,5) = -1(1,1,1)+2(1,2,3).

Both methods are correct.

Answer 17

would one be easier than another?

Answer 18

usually getting a matrix and row reducing (rref) can be a pain if done by hand. but it always works and is kinda mindless; just follow the set procedure. The other method usually involves thinking for a little, and sometimes you can get lucky and notice the linear combination. But sometimes its not so clear and the linear combination might not be obvious.

I usually stare at the problem for a minute trying to guess the linear combination, and if I cant, then I create a matrix and rref, etc.