For A B C D answer for each question based on the properties

For A B C D answer for each question based on the properties given. Linear algebra

Let V be a v.s. of dimension n > 0 and let S = {u_1, u_2, ..., u_m} be a collection of vectors of V. Based on this information answer the following questions. If your answer is yes, you must provide a brief explanation to support your answer. If your answer is no, you must provide a counterexample to support your answer. If m > n, can we conclude that S is a spanning set for V? If m > n, can we conclude that S is a linearly dependent set? If m

Solution

a.

False. S could be a collection of zero vectors which all in V. Hence they would not span V as n>0

b.

Yes.

A dimension of a finite vector space is given by the largest possible set of linearly independent vectors.

So if m>n and S is linearly independent then dimension of V would be m>n which is not possible

c.

False.

All vectors of S could be zero vectors

d.

False

All vectors of S could be 0 vectors hence linearly dependent.