let v1 and v2 span a vectorspace V. Let v3 be any other vector in V. Show that v1,v2 and v3 also span V.
Solution
Let w be in V, then w=av1+bv2 since V=span({v1,v2}), for some a, b scalars. Therefore, w=av1+bv2+0v3, i.e., w is in span({v1,v2,v3}), which since w is arbitrary means that V is contained in span({v1,v2,v3}). Now, notice that since v3 is in V and V=span({v1,v2}), then v3=xv1+yv2 for some x, y scalars. Now, let w be in span({v1,v2,v3}), then w=av1+bv2+cv3 for some scalars a, b and c. Then w=av1+bv2+c(xv1+yv2)=(a+cx)v1+(b+cy)v2. Thus w is in span({v1,v2})=V, which since w is arbitrary means that span({v1,v2,v3}) is contained in V. Therefore, V=span({v1,v2,v3}).
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let v1 and v2 span a vectorspace V- Let v3 be any other vector in V- S.docx
1. let v1 and v2 span a vectorspace V. Let v3 be any other vector in V. Show that v1,v2 and v3 also
span V.
Solution
Let w be in V, then w=av1+bv2 since V=span({v1,v2}), for some a, b scalars. Therefore,
w=av1+bv2+0v3, i.e., w is in span({v1,v2,v3}), which since w is arbitrary means that V is
contained in span({v1,v2,v3}). Now, notice that since v3 is in V and V=span({v1,v2}), then
v3=xv1+yv2 for some x, y scalars. Now, let w be in span({v1,v2,v3}), then w=av1+bv2+cv3 for
some scalars a, b and c. Then w=av1+bv2+c(xv1+yv2)=(a+cx)v1+(b+cy)v2. Thus w is in
span({v1,v2})=V, which since w is arbitrary means that span({v1,v2,v3}) is contained in V.
Therefore, V=span({v1,v2,v3}).
