# Thread: Basis in Linear Algebra

1. ## 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.

2. 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, $\displaystyle \bold{i},\bold{j},\bold{k}$ 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.

3. 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

4. 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.

5. i think, TPH missed an important point.. the set must be linearly independent..

6. Originally Posted by kalagota
i think, TPH missed an important point.. the set must be linearly independent..
No! I said the representation is unique.

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

you already included a theorem in the definition..