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 ?
Final answer: Since
     
 
2
0
0
−3
−2  4  − 3 −1 + 5 = 0 ,
1
5
0
1
the set is linearly dependent. By homework problem LA 3.12, the span is not R3 . By inspection, the vectors
are not collinear so the span is not a line. Hence the span must be a plane.
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 ?
Final answer: The set is linearly independent. By homework problem LA 3.12, the span is R3 .

    !
3
−2
4
Problem 3. Is span −2 ,  2  , −1 a line, a plane, or R3 ?
1
−1
3
Final answer: The set is linearly independent. By homework problem LA 3.12, the span is R3 .
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.
Full solution: Let {v1 = ~0, v2 , . . . , vk } be an arbitrary set of k vectors in Rn that contains the zero vector.
Note that
1v1 + 0v2 + 0v2 + · · · + 0vk = ~0.
Since the first coefficient is nonzero, this shows that the set {v1 = ~0, v2 , . . . , vk } is linearly dependent. Hence,
there does not exist a linearly independent set that contains the zero vector.
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 }.
Full solution: The statement is true. If {v1 , v2 , v3 , v4 } is linearly dependent, then there exist c1 , c2 , c3 , c4 ∈
R, not all zero, such that
c1 v1 + · · · + c4 v4 = ~0.
Therefore, note that
c1 v1 + · · · + c4 v4 + 0v5 = ~0.
Since not all of the coefficients c1 , c2 , c3 , c4 and 0 are equal to 0, this shows that the set {v1 , v2 , v3 , v4 , v5 } is
linearly dependent.
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 }.
Full solution: This is false. Here is a counterexample. Consider the linearly independent set
 
 
 
 
1
0
0
0 )
(
0
1
0
0







v1 =   , v2 =   , v3 =   , v4 =  
0
0
1
0
0
0
0
1
in R4 (any other linearly independent set of 4 vectors in Rn will work just fine too). Let v5 = ~0. Note that
by Problem 4 above, the set {v1 , v2 , v3 , v4 , v5 } is not linearly independent because it contains the zero vector.
3