# Linear independancy/dependancy

• Nx2
In summary, Guppy is correct and you are incorrect. Any two vectors in a three-dimensional space can be considered coplanar.
Nx2
hi guys just want to clarify something...

If we are givin (vectors w u and v) w=au +bv what can we say about w
a) if u and v are independant?
b) if u an v are dependant?

this is what i got so far.
a) w cannot be writtin as a scalar multiple of u and v and are therefore not coplanar.
b)w can be written as a scalar multiple of u and v and are therefor coplanar.

but i think there is more to it. I don't think my answer explains enough.

in a) can i say that w=0?... just trying to find more properties of w for a) and b)... any help would be appreciated. thnks.

- Tu

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No, w is coplanar with u and v by virtue of being a linear combination of them. What is true is that if u and v are independent, then w is not a multiple of one or the other and is not "co-linear" with either. If u and v are dependent, then one is a multiple of the other, w is a multiple of each, and w points in the same direction as both u and v (which are "co-linear" since they are dependent).

We should know that any two vectors in a N-Dimensional space where N>1 should be coplanar since Plane is 2-dimensional , that is, any of the 2 vectors should lie on the same plane.

Guppy said:
We should know that any two vectors in a N-Dimensional space where N>1 should be coplanar since Plane is 2-dimensional , that is, any of the 2 vectors should lie on the same plane.

what? I am pretty sure that's wrong dude, what about two vectors in three space? your thinking is limited to 2 dimensions or 2-space..BTW Nx2, does that happen to be in the Harcourt mathematics 12 geometry and discrete math textbook?

In order to define a plane (through the origin), you need two (linearly independent) vectors. So it doesn't make much sense to talk about two vectors being coplanar. Sure, they're always both in some plane together, though, specifically, the plane that they define (if they're independent)!

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MiniTank said:
what? I am pretty sure that's wrong dude, what about two vectors in three space? your thinking is limited to 2 dimensions or 2-space..BTW Nx2, does that happen to be in the Harcourt mathematics 12 geometry and discrete math textbook?

No, Guppy is completely correct and you are completely wrong. Any two vectors define either a line (if one is a multiple of the other) or a plane (if one is NOT a multiple of the other).

Questions of that form appear in numerous texts, MiniTank, though it would be odd if you were right.

HallsofIvy said:
No, Guppy is completely correct and you are completely wrong. Any two vectors define either a line (if one is a multiple of the other) or a plane (if one is NOT a multiple of the other).

## What is linear independence?

Linear independence refers to the property of a set of vectors in a vector space where no vector can be expressed as a linear combination of the other vectors in the set.

## What is linear dependence?

Linear dependence refers to the property of a set of vectors in a vector space where at least one vector can be expressed as a linear combination of the other vectors in the set.

## How is linear independence/dependence determined?

Linear independence/dependence can be determined by performing Gaussian elimination on the set of vectors and checking if the resulting reduced row echelon form contains any zero rows.

## Why is linear independence/dependence important?

Linear independence/dependence is important in linear algebra because it helps determine the dimension of vector spaces, the number of solutions to systems of linear equations, and the basis of a vector space.

## Can a set of vectors be both linearly independent and dependent?

No, a set of vectors cannot be both linearly independent and dependent. It is either one or the other, as these properties are mutually exclusive.

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