Page 380 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition
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8. , , and , form a linearly dependent set in .
(a) Show that the vectors
(b) Express each vector as a linear combination of the other two.
For which real values of do the following vectors form a linearly dependent set in ?
9.
Show that if is a linearly independent set of vectors, then so are , , , , ,
is a linearly independent set of vectors, then so is every nonempty subset of S.
10. and .
Show that if
11.
Show that if is a linearly dependent set of vectors in a vector space V, and is any vector in V, then
12. is also linearly dependent.
Show that if is a linearly dependent set of vectors in a vector space V, and if are any vectors in
13. V, then is also linearly dependent.
Show that every set with more than three vectors from is linearly dependent.
14.
Show that if is linearly independent and does not lie in span , then is linearly independent.
15.
Prove: For any vectors u, v, and w, the vectors , , and form a linearly dependent set.
16.
Prove: The space spanned by two vectors in is a line through the origin, a plane through the origin, or the origin itself.
17.
Under what conditions is a set with one vector linearly independent?
18.
Are the vectors , , and in part (a) of the accompanying figure linearly independent? What about those in part (b)?
19. Explain.
