# Basis in Linear Algebra

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• Dec 9th 2007, 05:08 PM
akhayoon
Basis in Linear Algebra
what exactly is a basis?

all these textbook explantions are so confusing to me and usually have all kinds of symbols that I have to decipher making it even harder to read let alone understand.
• Dec 9th 2007, 05:32 PM
ThePerfectHacker
Quote:

Originally Posted by akhayoon
what exactly is a basis?
.

Let R be the real numbers and V be a vector space over R. The set of vectors {v1,v2,...,vn} is said to form a basis when any vector in V can be expressed* in terms of v_1,v_2,...,v_n AND in only one way.

For example, $\bold{i},\bold{j},\bold{k}$ is a basis for $V=\mathbb{R}^3$ over $\mathbb{R}$. Because anything in $V$ can be expressed in terms of those.

*)By "expressed" I mean as a linear combination of those vectors.
• Dec 9th 2007, 05:36 PM
akhayoon
but I thought if they were linear combinations that means they are linearly dependent and therefore are not a basis of that subplace or whatever they call it...lol
• Dec 10th 2007, 04:40 AM
badgerigar
Quote:

but I thought if they were linear combinations that means they are linearly dependent and therefore are not a basis of that subplace or whatever they call it...lol
If any vectors in a set are linear combinations of each other then they are linearly dependent and not a basis.
If a set is a basis of a space then every vector in the space is a linear combination of the vectors in the basis.
• Dec 10th 2007, 07:00 AM
kalagota
i think, TPH missed an important point.. the set must be linearly independent..
• Dec 10th 2007, 11:30 AM
ThePerfectHacker
Quote:

Originally Posted by kalagota
i think, TPH missed an important point.. the set must be linearly independent..

No! I said the representation is unique.
• Dec 11th 2007, 12:48 AM
kalagota
Quote:

Originally Posted by ThePerfectHacker
No! I said the representation is unique.

hehehe. Ü

you already included a theorem in the definition.. :)