Let v1→= and v1→= what is the **span** of the vector space defined by v1→...
Let and what is the **span** of the vector space defined by and ? Explain your answer in detail?
Answer & Explanation
Typically we talk about the span of a set of vectors, rather than of an entire vector space. We will proceed, then, in examining the span of within a given vector space.
The span of a set of vectors in a vector space is the set of all finite linear combinations of those vectors. That is, given a subset S of a vector space over a field F, we have
(the set of any finite sum with each term being the product of a scalar and an element of S)
For simplicity, we will assume that our given vector space is over some subfield F of . Then, applying the above definition:
But note that , and so, for any
Then, as any linear combination of can be expressed as a scalar multiple of , and any scalar multiple of can be expressed as a linear combination of by setting we have