How can I prove that the space spanned by two vectors in R^n is a line through the origin, a plane through the origin, or the origin itself?

Question

turingtest · Answer

i'm very rusty on LA, but let's consider what we need to show that a given set of vectors spans a vector space: that every vector in that space can be written as a linear combination of the given vectors
the simplest case is that which only spans the origin. what vectors would only span the origin?

anonymous · Answer

it would be (0,0)

anonymous · Answer

or a zero vector

turingtest · Answer

right, and we can see this by seeing that$$c_1\langle0,0\rangle+c_2\langle0,0\rangle=\vec 0$$now what kind of vectors would span $\mathbb R^1$ ?

anonymous · Answer

it would be (0,1) or (1,0)

turingtest · Answer

well you asked for two vectors, and the don't have to be unit vectors
so if we are to choose two vectors that together only span $\mathbb R^1$ they must have a certain relation... which is what?

anonymous · Answer

linearly dependent?

turingtest · Answer

yep :) so we can write this as$$\vec v_1=\langle x,y\rangle\\vec v_2=\langle c x, cy\rangle$$if you take the determinant of this you will find it's zero, which shows it doesn't span $\mathbb R^2$

turingtest · Answer

it's easy to show that either one individually spans $\mathbb R^1$ though
as for vectors that span $\vec R^2$ what property must they have?

turingtest · Answer

$\mathbb R^2$*

anonymous · Answer

it must be closed under addition and scalar multiplication

turingtest · Answer

well do you know how you would show that for the last case?

anonymous · Answer

i guess by multiplying it by a scalar, k where k=/= 0?

anonymous · Answer

@amistre64 any help here?