Question

i need some explain about the matrix change of basis cuz i want understand it deeply , so anyone help me ?

Answer 1

@amistre64

Answer 2

@ikram002p

Answer 3

@Nnesha

Answer 4

@dan815

Answer 5

@IrishBoy123 its not sweet nut now

Answer 6

Explain what you know or think about change of basis currently.

Answer 7

@Empty i mean when i have Linear map we use matrix for change of basis , u study this thing in algebra my mind breaks down when i see it , and i dont know why i study this thing

Answer 8

Oh ok, well if that's the case then I can explain it to you. :)

Answer 9

@Empty ty

Answer 10

@nincompoop

Answer 11

@Jamierox4ev3r

Answer 12

So here the vectors are not the unit vectors! They are:
\(\vec a = \langle 2,0\rangle\) and \(\vec b = \langle 0,2\rangle\)
So when I write that combination of vectors \(\vec v = \langle 3, 1\rangle \) that means with respect to this basis! Written out a little more: \(\vec v = 3\vec a + 1\vec b\). So really this is shorthand for a linear combination of our basis vectors.

But let's think about this, if we are looking at where to drill holes on a square sheet of metal, that location, vector \(\vec v\) is NOT going to be \( \langle 3, 1\rangle \). What we need to respect is that the vector \(\vec v\) is pointing to an invariant location in space! The numbers are just what we use to do calculation and help us program our drilling machine for example. So due to the precision of drilling, the machine might have a different coordinate system than the one I used with \(\vec a\) and \(\vec b\).

So we can have a new coordinate system for the computer that manages the drilling machine that looks like this: So here we have the same point, but represented with a different basis. Since we know \(3 \vec a + 1 \vec b = 9 \vec x + 3 \vec y\) this is one kind of made up example where we can solve for these basis vectors.

Really this is a simplified example but I hope it kinda gives the idea of why this might be useful. Try solving for \(\vec x\) or \(\vec y\) maybe or ask any kind of clarifying questions I can try to help! :D