some linear algebra questions?

[korean]

i have some question that im not too sure of....i have got an answer but im not sure or i need someone to just point me in the right direction. thanks!


1. two vectors are linearly dependent if and only if they lie on a line through the origin?
my random guess is true but im not sure?... my random thought is the two vectors can be multiples of another? but that doesnt make much since to me either.

2. if x and y are linearly independent and if {x,y,z} is linearly dependent, then z is in span {x,y}
my first answer is true. but when i think about it not every vector in a linearly dependent set is a linear combination. (I THINK) so this is why i am not sure if im right. if i am right i think the answer is false.

3. (almost the same as 2) if x and y are linearly independent, and if z is in span {x,y}, then {x,y,z} is linearly dependent?
my answer is true. characterization of linearly dependent sets theorem. (theorem 7 in my book)

4. if v1,v2,v3,v4 are in R4 and v3 is not a linear combination of v1,v2,v4, then {v1,v2,v3,v4} is linearly independent.
this one im just lost if anyone can point me in a right direction im sure i can work on it and see what i get. my first reaction is to say false but i have no justification.

 

[stapel]

1) You might want to think visually, using the special (and very simple) case of the xy-plane. According to your answer, the unit vector for the x-axis and the unit vector for the y-axis are dependent. But are they? Don't they actually form a spanning set?

2) I'm not sure why you think your answer is wrong, since the reasoning you give justifies your answer...?

3) Sounds good to me.

4) For {v₁, v₂, v₃, v₄} to be linearly independent, none of them must be a linear combination of any of the others. But (hint) the fact that v₃ is not a linear combination of the other three vectors does not mean that, say, v₁ is not a linear combination of v₂ and v₄.

Hope that helps!

Eliz.