@user85795 :-) that sort of sh*t always goes on. like a boxer represents an entire nationality :-)

ref should have let Usyk take a few more shots instead of finishing on points.

Well, in the first case, your solutions are complex. In the second, they are real.

Yeah I realized you use the relation $a+bi = c + di \implies a = c, b = d$

so the n equation system becomes a 2n equation system I think

So in a way we're "ditching" the imaginary parts?

I mean, you can do things that way, I suppose.

If your goal is just computation.

you're implicitly defaulting to counting real equations, which isn't something you really have to do, but yes

i don't know what you mean by "ditching" imaginary parts

Yeah it's weird, I'm just reviewing some linear algebra and this resource does that. It's strange because you're taking the complex part of the number and shedding the $i$

if z and w are complex numbers, to say "z = w" is the same thing as saying the pair of things "Re(z) = Re(w) and Im(z) = Im(w)"

you might prefer the latter if for some reason you are more comfy with real numbers than complex numbers, but it isn't as though whenever anybody writes the first thing, they are "really" writing the second, or that your mind otherwise really needs to think of the second thing as simpler or better or more fundamental than the first

Ah okay so it's just a preference thing

You can solve it over $\mathbb{C}$ and then do the Re(z) = Re(w) ,etc

it might be that in some programming contexts, you have a data type that is roughly 'real number' and when you write out + and * without specifying the types of these operations, something in the language or the interpreter/compiler will default to assuming that, and you need to do something 'more' (e.g. define your own data type) to get complex * and + as first-class operations

obliv: or never "do the Re(z) = Re(w), etc."

a big part of linear algebra at one level of abstraction is realizing how much of it works over any field. row reducing a matrix involves adding rows to other rows, and scaling them by nonzero scalars. "scaling" if it's complex numbers does involve complex multiplication, but from the point of view of the row operation, you really don't need to think in terms of "i have to do (a+bi)(c+di) = (ac - bd) + (ac+bd)i, where the multiplication on the right hand side is the only true multiplication"

and it's maybe better if you don't think of it that way

maybe think of the formula for entries of A^{-1} in terms of the entries of A, when A is an invertible 2x2 matrix. exactly the same formulas in both the real and complex cases

it isn't like there's a "complex formula" that's somehow different from the "real formula". it's the same formula, and it only becomes complicated if you want a formula for the real and imaginary parts of the entries of A^{-1} in terms of the corresponding parts of the entries of A

and maybe you want that, but i don't know why you'd automatically want that as a consequence just of wanting to do linear algebra over C

Well i've actually never solved systems of linear equations in anything but $\mathbb{R}$ so this abstraction is new to me. I'm actually pretty curious about this subject now that I'm reviewing it

And eventually I'll want to learn more about roots of nonlinear univariate polynomials and solutions to systems of such polynomials

over various fields

@Obliv $p$-adics it bust, baby!

like linear algebra is the study of systems of linear multivariate equations but I wonder how much more difficult it becomes when you allow the equations to be nonlinear

Or even systems of equations that aren't just polynomials..

@Obliv linear algebra is not, fundamentally, about solving linear systems of equations. That is just a thing you can do with linear algebra, or a model for thinking about things.

Linear algebra is more about they study of abstract vector spaces.

I thought that was abstract algebra

I have got cat hair all/me.

I don't get why abstract algebra isn't taught first. Module theory precedes vector spaces

lint roller is your friend @koro

this cat is near me almost whole day :)

@Koro eat it

bruh

and it doesn't scratch me when I lift it and play with it :).

I have probably ate so much unintentional cat hair in my life when i used to have a cat

I still have my scars as memories from my cat ;) I miss him

@Obliv "Abstract algebra" is a course title. It usually includes some group theory, some ring theory, and some stuff about modules and vector spaces.

@Obliv OR once you understand the relatively simple case of a vector space over a field, you can start to look at modules over rings.

More accurately, abstract algebra can refer to a course title

Lizard :)

I thought that was a fox for a second

What a qt

he's about 4 months now, already abandoned by his mother who visits here often but the frequency of her visits has reduced a lot since she abandoned the kittens (including this one).

@Obliv he is very furry and his legs are short and tail looks like cotton candy sometimes :)

a hairy cotton candy

:)

I think as he grows he realises the strength of his paws.

in past, I've been by scratched consecutively for days by him and his sister both.

but not now

koro: cute

that's his big brother.

this is how the kitten will look like perhaps in about 3 to 4 months from now.

it's looking into the camera so perfectly. 😅

@Koro big brother is watching you

@SoumikMukherjee he's surprisingly photogenic.

yeah he is looking directly to the camera

👀