Question

Is anyone good at linear algebra that could help me understand some concepts at a deeper level then just a definition?

Answer 1

do you want help in groups?

Answer 2

else what topics in li. algebra?

Answer 3

vector spaces...........?

Answer 4

what do you mean in groups? so far we have gotten through vector spaces, subspaces, span, linear independence
, and now we are on to linear functions(maps?)

Answer 5

and i understand vector spaces,subspaces, and linear independence but what exactly is the difference between span and linear independence?

Answer 6

anyone know?

Answer 7

Given a set of vectors v_1,...,v_n, a linear combination of those vectors is a vector of the form a_1v_1+...+a_nv_n, for a_1,...,a_n some scalars. The span of the vectors v_1,...,v_n is the set of all the linear combinations of those vectors. I.e., it is the set whose elements are the vectors of the form a_1v_1+...+a_nv_n for all possible values of the coefficients a_1,...,a_n.

Answer 8

ok so the SPAN tells us the we could get any set of vectors in R^n? where linear indepencence just tells us if there are any simmilar vectors?

Answer 9

linearly independent vectors of \(\mathbb R^n\) will span a space
this is not necessarily all of \(\mathbb R^n\)

Answer 10

ok so what is the point of span?
why is it signifigant?

Answer 11

i think understand why linear independence is important

Answer 12

say your in \(\mathbb R^3\), and only two of your vectors are linearly independent, (ie one in a scale multiple of another, ),
your span can only be single plane in \(\mathbb R^3\)

Answer 13

ohhhh ok i see where if 3 were linearly independent the span would be all of R^3? (sorry i dont know how to type the facny R lol)

Answer 14

\[\mathbb{R} ^{3}\]

Answer 15

thats right , you could get any where in the solid \ (\mathbb R^3\), by taking a combination of steps in each of the three directions,

Answer 16

Ah ok i understand! ok so from that we went to Dim's, Bases, and Rank. the only thing im unsure about is what does the basis tell us? i know how to get it and how to get the dimension and rank but not exactly what it means

Answer 17

if all three vectors were a multiple of each other you could only move along a line , taking combinations of three different sized steps

Answer 18

thats why you can cut out the other vectors that are linearly dependent right? becasue they are not contributing any new information

Answer 19

yeah,
they will only have scale information,