Math 3321 – Lecture 20 notes
5.7 Vectors; Linear Dependence and Linear Independence
Terms:
Row Vectors –
Column Vectors –
n-tuples –
Vector operations:
Linear combination of vectors:
Linear Dependence and Linear Independence
u and v are linearly dependent if one of the vectors is a scalar multiple of the other and
they are linearly independent if neither is a scalar multiple of the other.
The set of vectors {v1, v2, . . ., vk} in Rn is linearly dependent if there exist k numbers c1,
c2, . . ., ck, not all zero, such that c1v1 + c2 v 2 +L+ ck v k = 0
The set of vectors {v1, v2, . . ., vk} is linearly independent if c1v1 + c2 v 2 +L+ ck v k = 0
implies c1 = c2 = L = ck = 0
Let {v1,v 2 ,v 3 ,...., v k } be k vectors in
• If k = 2 then just check to see if the vectors are scalar multiples of each other
• If k = n: compute
• If
k≤n
solve



det 



Ac = 0 :


 = 0 linearly dependent

→
≠ 0 linearly independent

| 

| 
.. |
0 
.. |
0 
.. v k 0 
0 
.. |

0 
.. |
|
| .. .. |
|
| .. .. |
v1 v 2 .. .. v k
|
|







|
|
.. ..
.. ..
|
|
|
|
v1 v 2
|
|
|
|
..
..
..
..
..
If you have a non-trivial solutions (not all c’s are zero) then linearly dependent
If you have only trivial solutions, then linearly independent
• If k > n => linearly dependent
If we are checking for linear dependence/independence for functions, we can use the
Wronskian. If W(x) ≠ 0 for at least one x on an interval then the functions are linearly
independent on that interval.
Examples:
1. Determine whether the set of vectors is linearly dependent or linearly independent. If it
is linearly dependent, express one of the vectors as a linear combination of the others.
a) {( 1, 2, 5 ), ( 1, −2, 1 ), ( 2, 1, 4 )}
b) {( 0, 0, 0, 1) ( 4, −2, 0 2), ( 2, −1, 0 1), ( 1, 1, 0 1)}
c) v1 = (1,0,2,−2), v 2 = (2,1,0,1), v 3 = (2,−1,0,1)
2. For which values of b are the vectors ( 2, −b ), ( 2b + 6, 4b ) linearly dependent?
3. Calculate the Wronskian of the set of functions. Then determine whether the functions
are linearly dependent or linearly independent:
f1 (x) = x, f2 (x) = x 2 , f3 (x) = x 3; x ∈(−∞,∞)
5.8 Eigenvalues and Eigenvectors
Defn: Let A be an nxn matrix. A number λ is an eigenvalue of A if there exists a
nonzero vector v in
such that Av = λ v . The vector v is called the eigenvector
corresponding to λ .
λ is an eigenvalue of A if and only if det(A − λIn) = 0
Examples:
1. Find the eigenvalues of
 1 −3 
A=
 −2 2 
 2 1 1 


2. Find the eigenvalues of A =  1 2 1 
 −2 −2 −1 
Finding eigenvectors:
To find the eigenvectors, we need to find the non-trivial solution of (A − λ In)x = 0
Examples:
 1 −3 
1 (continued) Find the eigenvectors of A =  −2 2 


2 (continued) Find the eigenvectors of
 2 1 1
A= 1 2 1

 −2 −2 −1




More examples:
3) Find the eigenvalues and eigenvectors of
 2 −1 
A=
 1 2 
4) Find the eigenvalues and eigenvectors of
 0 3 
A=
 −4 7 
5) Find the eigenvalues and number of independent eigenvectors. (Hint: -1 is an
eigenvalue.)