3.1: 11, 16, 17
11. Prove that any finite set of vectors that contains the zero vector must be
linearly dependent.
Proof. If {v1 , v2 , . . . , vn } is a finite set of vectors and one of them is the zero
vector, without loss of generality we may assume that v1 = 0. Then with
constants c1 = 1 and c2 = · · · = cn = 0, we have that
c1 v1 + c2 v2 + · · · + cn + vn = 1 · 0 + 0 · v2 + · · · + 0 · vn = 0.
Therefore, any finite set of vectors that contains the zero vector must be linearly
independent.
16. Let {v1 , . . . , vn } be a spanning set for the vector space V , and let v be any
other vector in V . Show that v, v1 , . . . , vn are linearly dependent.
Proof. Since {v1 , . . . , vn } is a spanning set for V , every vector in V can be
written as a linear combination of v1 , . . . , vn . Thus v = c1 v1 + · · · cn vn for
some scalars c1 , . . . , cn . But then
(−1)v + c1 v1 + · · · cn vn = 0,
and thus v, v1 , . . . , vn are linearly dependent.
17. Let v1 , v2 , . . . vn be linearly independent vectors in a vector space V . Show
that v2 , . . . , vn cannot span V .
Proof. Since v1 , v2 , . . . vn are linearly independent vectors in V , if
c1 v1 + c2 v2 + · · · cn vn = 0
then all the scalars c1 , . . . , cn must equal 0. Thus there is no way to write
v1 = c2 v2 + · · · cn vn .
In other words, v1 cannot be written as a linear combination of v2 , . . . , vn .
Therefore, v2 , . . . , vn cannot span V .
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