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?

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Julia Adams · Accepted Answer

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

Hazel Anglin · Answer

undefined The span of the vector space defined by vec(v_1) and vec(v_2) is the set of all possible linear combinations of these vectors. In this case, since vec(v_1) = [2, 3] and vec(v_2) = [4, 6], any vector that can be obtained by multiplying vec(v_1) and vec(v_2) by scalar values and adding them together will be in the span of the vector space.

To find the span, we need to determine all possible linear combinations of vec(v_1) and vec(v_2). Since vec(v_2) = 2 * vec(v_1), vec(v_2) is a scalar multiple of vec(v_1). Therefore, the span of the vector space defined by vec(v_1) and vec(v_2) is the set of all possible scalar multiples of vec(v_1), which forms a line passing through the origin and vec(v_1).

In summary, the span of the vector space defined by vec(v_1) and vec(v_2) is a line passing through the origin and vec(v_1), since vec(v_2) is a scalar multiple of vec(v_1).