MATH 51 SECTION 2, THURSDAY 1/6/2011
Henry Adams
henrya@math.stanford.edu
Office 380N
Office hours: For this week only, Thurs 2:15-3:45 and Fri 12:30-2:00
Course website: http://math51.stanford.edu
1. Review of main concepts
Let v1 , . . . , vk be vectors in Rn .
• A vector of the form
c1 v1 + c2 v2 + · · · + ck vk ,
where c1 , . . . , ck ∈ R, is a linear combination of {v1 , . . . , vk }.
• The span of {v1 , . . . , vk } is the set of all linear combinations of {v1 , . . . , vk }. That is,
span(v1 , . . . , vk ) = {c1 v1 + · · · + ck vk | c1 , . . . , ck ∈ R}.
• The set {v1 , . . . , vk } is linearly dependent when
c1 v1 + · · · + ck vk = 0
for some c1 , . . . , ck ∈ R not all of which are zero.
• Conversely, the set {v1 , . . . , vk } is linearly independent if whenever
c1 v1 + · · · + ck vk = 0,
we have that c1 = · · · = ck = 0.
2. Problems
(     )
−3
2
0
Problem 1. Consider the set of vectors  4  , −1 , 5 . Show that this set is linearly (in)dependent,
1
1
5
whichever the case may be. Is the span of this set a line, a plane, or R3 ?
1
(     )
2
−2
3
Problem 2. Consider the set of vectors 2 , −4 , −1 . Show that this set is linearly (in)dependent,
1
−1
1
whichever the case may be. Is the span of this set a line, a plane, or R3 ?

    !
3
−2
4
Problem 3. Is span −2 ,  2  , −1 a line, a plane, or R3 ?
1
−1
3
2
Problem 4. Does there exist a linearly independent set that contains the zero vector? If so, give an example.
If not, explain why not.
Problem 5. Show that the following statement is true, or show that it’s false by finding a counterexample:
Suppose v1 , . . . , v5 ∈ Rn . If {v1 , v2 , v3 , v4 } is linearly dependent, then so is {v1 , v2 , v3 , v4 , v5 }.
Problem 6. Show that the following statement is true, or show that it’s false by finding a counterexample:
Suppose v1 , . . . , v5 ∈ Rn . If {v1 , v2 , v3 , v4 } is linearly independent, then so is {v1 , v2 , v3 , v4 , v5 }.
3