Question

Let `vec(v_1) = [( 2),(3)]` and `vec(v_1) = [( 4),(6)]` what is the **span** of the vector space defined by `vec(v_1) and vec(v_1)`? Explain your answer in detail?

Answer 1

`"span"({vecv_1,vecv_2}) = {lambdavecv_1|lambdainF}`

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 `{vecv_1,vecv_2}` 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
`"span"(S)={sum_(i=1)^klambda_ks_k|ninNN,s_iinS,lambda_iinF}`

(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 `CC`. Then, applying the above definition:

`"span"({vecv_1,vecv_2}) = {sum_(i=1)^2lambda_iv_i|lambda_iinF}`

`= {lambda_1vecv_1+lambda_2vecv_2|lambda_1,lambda_2inF}`

But note that `vecv_2 = 2vecv_1`, and so, for any `lambda_1,lambda_2inF`,

`lambda_1vecv_1+lambda_2vecv_2=lambda_1vecv_1+lambda_2(2vecv_1)=(lambda_1+2lambda_2)vecv_1`

Then, as any linear combination of `vecv_1` and `vecv_2` can be expressed as a scalar multiple of `vecv_1`, and any scalar multiple of `vecv_1` can be expressed as a linear combination of `vecv_1` and `vecv_2` by setting `lambda_2=0`, we have

`"span"({vecv_1,vecv_2}) = {lambdavecv_1|lambdainF}`