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?

1 Answer
Jun 7, 2016

"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}