Thread: 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.

Originally Posted by akhayoon

what exactly is a basis?
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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, is a basis for over . Because anything in can be expressed in terms of those.


*)By "expressed" I mean as a linear combination of those vectors.

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

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.

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

Originally Posted by kalagota

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

No! I said the representation is unique.

Originally Posted by ThePerfectHacker

No! I said the representation is unique.

hehehe. Ü

you already included a theorem in the definition..