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

Answer:

#"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}#