I can't figure out how to solve this system of equations. what am I doing wrong?

[rose jasper]

The best I can do with GE is eliminate one line to zeroes, and then if I try to back substitute that into the original problem, I get the wrong answer!

[willow hearth]

"4x1 = x2" does not mean "any vector whose first entry is 4 times the second"

Think carefully

[rose jasper]

haaa, interesting. I guess it means "x_2 is four times greater than x_1" which would be [1 4] 😅

thank you!

[rose jasper]

so, this problem still confuses me, because I would have absolutely no idea how to re-use this same logic in R^3. For instance, what eigenvector would this (post GE) matrix imply??:

it seems to me there has to be a more "linear algebra"-ey way to find this solution eigenvector without backsubstitution followed by some ad hoc algebraic interpretation

[willow harbor]

You have to define the parameters first.

In the firs case, the parameter is x1, and x2 depends on x1

let's write x1 = t, then x2 = 4t

So, in the firs position goes t and in the second, goes 4t

Thus, the vector space can be writen as (t, 4t) = t*(1, 4)

is a space generated by the vector (1; 4)

In the second case you have a 2-dimensions vector space

I can chose x1 = t and x2 = u

and x3 = -4t - u

so, the vector space can be represented as (t; u; -4t - u) = t*(1; 0; -4) + u*(0; 1; -1)

So, any vector in this vector space can be written as a linear combinations of the vectors (1; 0; -4) and (0 ;1; -1). These are the generators of the vector space

[rose jasper]

ohhh my gosh that's... a really interesting way to establish that. let me think on this for a while

[willow hearth]

t(1; 0; -4) + u(0; 1; -1)

[willow harbor]

Yes! Thanks for the correction. I edited already. Thank you very much 🙂

[gritty skiff]

@willow harbor has given 1 rep to @willow hearth

[willow harbor]

This concept is key in linear algebra courses. I assume you will start to see it very soon🙂

[rose jasper]

yeah, I've actually taken a few linear algebra courses now and for some reason non of them have introduced this framing. I've seen two different flavors of linear algebra course material: 1) do trivial examples which the numbers line up such that you can solve them in your head and 2) figure out the general rules such that a programmatic solution can be applied.

and in neither of those paradigms is this framing you're discussing here approached

[willow harbor]

I taught in many linear algebra courses at the university and in every of each we saw vector spaces. Maybe I generalized too fast. If you don't see it, nevermind.

[rose jasper]

I definitely have an understanding of matrix spaces (and how they are composed of vector spaces) from the classes I took, but this numerical approach of dummy variables in analyzing those spaces was never brought up