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.

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- Dec 9th 2007, 04:08 PMakhayoonBasis 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, 04:32 PMThePerfectHacker
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, $\displaystyle \bold{i},\bold{j},\bold{k}$ is a basis for $\displaystyle V=\mathbb{R}^3$ over $\displaystyle \mathbb{R}$. Because anything in $\displaystyle V$ can be expressed in terms of those.

*)By "expressed" I mean as a linear combination of those vectors. - Dec 9th 2007, 04:36 PMakhayoon
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, 03:40 AMbadgerigarQuote:

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 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, 06:00 AMkalagota
i think, TPH missed an important point.. the set must be

**linearly independent..** - Dec 10th 2007, 10:30 AMThePerfectHacker
- Dec 10th 2007, 11:48 PMkalagota