Question

Spanning set & Linearly (In)dependent

Question below. I'm stuck. I don't know what's next...

Answer 1

2 vectors define a plane, a 2d object ... can a 2d object touch all of 3d space?

Answer 2

and independant set is one in which each vector cannot be defined in terms of other vectors in the set

Answer 3

no? 2d can't span all 3d space, so it is not a spanning set of R^3?

so since c2 needs c1, then it is linearly dependent?

Answer 4

since 2,4,0 is not a scaled version of 2,1,3 the vectors are independant.

Answer 5

one way to consider it is ...

a v1 + b v2 = v3 for all vectors (v3) in R^3

in order to span (to cross over, to reach all points) of R^3, you have to be able to create them from the given conditions ...

Answer 6

a(2,1,3) + b(2,4,0) = (x,y,z)

2a + 2b = x
a + 4b = y
3a + 0b = z

can the vector (0,1,3) be obtained?

2a + 2b = 0; b=-1
a + 4b = 1; 1-4 doest eqaul 1
3a + 0b = 3 ; a = 1

Answer 7

think of a piece of paper, you can draw 2 arrows on it that can be combined to reach all points in that paper ... but in order to get off of the paper you need to add a vector that is not contained on the paper ... a vector that points above/below it.

Answer 8

oh okay... i think i understand it better. two vectors = only spans a plane

so if in case i'm given three vectors, there will be a possibility that they can span R^3 right?

Answer 9

it requires n, independant vectors, to span R^n ... yes

Answer 10

... at least might need to be included for rigor

Answer 11

two independant vectors create a plane yes

Answer 12

for the linearly independent/dependent,

Answer 13

oh i think it makes sense to me now

Answer 14

correct enough yes

Answer 15

okay thank you again! :)

Answer 16

youre welcome