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.
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.
If any vectors in a set are linear combinations of each other then they are linearly dependent and not a basis.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.