# What does linear dependence (among vectors) actually mean geometrically?

Does it have any geometrical meaning? Also, I would ask is there any physical entity described by a linear combination and or linear dependent vectors?

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## 1 Answer

Geometrically, the span of a set $\{w_1\dots w_k\}$ is a subspace of the whole vector space. Picture a line through the origin, or a plane through the origin of 3-space.

If adding $v$ to a set $\{w_1\dots w_k\}$ results in a linearly dependent set, then that means that $v$ lies in the subspace generated by the $w_i$. In this sense, dependence indicates membership. Consider: adding 0 to any set makes the set linearly dependent, hence 0 is contained in all subspaces. If a set generates the whole space, then of course any vector added will be dependent, because it is guaranteed to belong to the span.

Every vector in a vector space is expressible with arbitrarily ridiculous combinations of dependent vectors. Say: $v=w-w+x-x+0+0+v$.

More interestingly, any vector is expressible as a unique linear combination of basis elements of a vector space. This uniqueness is both attractive and practical.

I guess this depends on what you intended to mean by "physical entity," but try this. Usually we think of unit vectors along orthogonal coordinate axes as "measuring sticks," and measurements of distance that we make in 3-space are linear combinations of these three rulers (which retain their sense of direction).

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the answer to the first question is what I was looking for, thank you very much..the idea of physical entity I was thinking about derives by the one of matrix..a matrix is made up of vectors right? So, is there a matrix or set of matrices (a set of linear combinations right?) able to describe a planet, or an atom, describe it in such a way I can reconstruct somehow everything I could possibly want to know about that planet or atom, perhaps except infos related to its interactions with other "physical objects".. –  Matteo Nov 29 '12 at 17:12
@Matteo A matrix is "made up of" vectors in the sense that its rows and columns are vectors, but I don't think I would describe the connection that way. The way to think of a matrix is: a realization of a transformation of the vector space, depending on a basis you have chosen. Transformations will be realized as different matrices depending on the basis you choose. You're right a vector makes linear combinations of the rows (or columns) of a matrix, depending on which side you're multiplying on. Linear transformations can be used to bundle up a lot of info about atoms, buildings, etc. –  rschwieb Nov 29 '12 at 17:25
@Matteo A prof once described a structure matrix used to evaluate the stability of a building. He said it was possible to evaluate the stability of the structure based on the eigenvalues of its matrix. –  rschwieb Nov 29 '12 at 17:27
What a wonderful subject this is –  Matteo Nov 29 '12 at 17:46
@Matteo Mathematical and physical sciences give other fields a lot to be jealous about :) –  rschwieb Nov 29 '12 at 17:59