You can determine special relationships among a set of vectors by deter- mining whether a linear combination of them can produce the zero vector.
In this lesson, you will learn how to
• recognize the identity matrix by notation and by form
• use row-reduced echelon form to find a linear combination of a vector in terms of a group of other vectors (if it exists)
• interpret linear dependency inR3 geometrically
Example 1
Problem. Write (0,0,0) as a linear combination of the given vectors.
a. A= (1,2,3), B= (4,5,6), C= (7,8,10) b. A= (1,2,3), B= (4,5,6), C= (7,8,9)
Solution. Using Lemma 3.1 from Lesson 3.3, construct an augmented matrix whose first three columns are the three given vectors, and whose fourth column is the zero vector. Then reduce the matrix to echelon form.
a.
⎛
⎝1 4 7 0 2 5 8 0 3 6 10 0
⎞
⎠ →
⎛
⎝1 0 0 0 0 1 0 0 0 0 1 0
⎞
⎠ →
a = 0 b = 0 c = 0
According to the echelon form, the only possible solution is to set all the scalars
to 0. ←−
The “all zero” solution is typically referred to as the
trivial solution. b.
⎛
⎝1 4 7 0 2 5 8 0 3 6 9 0
⎞
⎠ →
⎛
⎝1 0 −1 0
0 1 2 0
0 0 0 0
⎞
⎠ →
a−c = 0 b+ 2c = 0
For these vectors, there are nontrivial solutions: any combination of a, b, and c, wherea=c andb=−2c. One such possibility is
2(1,2,3)−4(4,5,6) + 2(7,8,9) = (0,0,0) which you can get by letting c= 2.
You can always find a linear combination of any set of vectors that will produceO: simply let all the scalars equal zero. But special relationships can be revealed if you can find nonzero combinations.
3.4 Linear Dependence and Independence
Definition
• Vectors A1, A2, . . . , Ak are linearly dependent if there are num-
bersc1, c2, . . . , ck that arenot all zero so that ←−
The key phrase in the definition isnot all zero, otherwise every set of vectors would be linearly dependent.
c1A1+c2A2+ã ã ã+ckAk=O whereO= (0,0, . . . ,0).
• On the other hand, the vectors are linearly independent if the only solution toc1A1+c2A2+ã ã ã+ckAk =O is c1 = c2 =ã ã ã = ck = 0.
The definitions of linearly dependent and linearly independent are stated in terms of algebra. But there is a geometric interpretation in R3: three vectors inR3 are linearly dependent if and only if the three vectors lie in
the same plane. So, for example, (7,8,9) is in the plane spanned by (1,2,3) ←−
You can extend this idea to any dimension, even if you cannot visualize it: “n vectors inRn are linearly dependent if and only if they all lie in the same hyperplane.” What does this mean inR2? and (4,5,6) but (7,8,10) is not. The next theorem states this in general.
Theorem 3.2
Vectors A1, A2, . . . , Ak are linearly dependent if and only if one of the vectors is a linear combination of the others.
For Discussion
1. Prove the statement “If vectorsA1, A2, . . . , Ak are linearly dependent, then one ←−
Since the statement in Theorem 3.2 is “if and only if,” the proof comes in two parts to prove each direction. This For Discussion problem proves one direction. You will prove the other direction in Exercise 9.
of the vectorsAiis a linear combination of the other vectors.”
To begin, you can say that if the vectors are linearly dependent, then by the definition of linear dependence,
c1A1+c2A2+ã ã ã+ckAk=O
for some set of numbersc1, c2, . . . , ck where at least one scalar is not zero. Letci
be the first nonzero among the scalars.
Complete this half of the proof by showing thatAican be written as a linear combination of the other vectors.
Example 2
Problem. Are the vectors (1,1,−1),(−1,1,0), and (2,1,1) linearly dependent or independent?
Solution. The vectors are linearly dependent ifa(1,1,−1) +b(−1,1,0) +c(2,1,1) = (0,0,0) has a nonzero solution. Set up an augmented matrix and reduce to echelon form.
⎧⎨
⎩
a−b+ 2c = 0 a+b+c = 0
−a+c = 0
→
⎛
⎝ 1 −1 2 0
1 1 1 0
−1 0 1 0
⎞
⎠
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Reducing that matrix to echelon form gives Remember You can construct the augmented matrix directly by writing the given vectors as the first columns, and then the desired linear combination vector as the last column.
⎛
⎝1 0 0 0 0 1 0 0 0 0 1 0
⎞
⎠
So the only solution is a = 0, b = 0, c = 0. Thus, these three vectors are linearly independent.
Example 3
Problem. Are the vectors (1,2,3),(4,5,6), and (7,8,9) linearly dependent or inde- pendent?
Solution. These vectors should be old friends by now. You could set up a matrix and reduce it to echelon form, but you’ve seen in Example 1 earlier in this lesson that
(7,8,9) = 2(4,5,6)−(1,2,3) So, the vectors are linearly dependent by Theorem 3.2.
←−Note that (1,2,3)− 2(4,5,6) + (7,8,9) = (0,0,0)
Here is a graph of the three linearly independent vectors from Example 2.
B C
A AA
The vectors (1,1,−1),(−1,1,0), and (2,1,1) do not lie in the same plane.
Here are two views of the three linearly dependent vectors from Example 3.
A B C
A B
C
These vectors lie in the same plane.
Facts and Notation
The elementary row operations will leave any column in a matrix consisting entirely of
zeros unchanged. So when testing linear dependence or independence, adding that final ←−
column of zeros provides no additional information—it will not change no matter what Why?
elementary row operations you take to reduce the matrix.
3.4 Linear Dependence and Independence
When a set of vectors inR3is linearly independent, the echelon form of the augmented matrix, which represents the only possible solution, is
⎛
⎝1 0 0 0 0 1 0 0 0 0 1 0
⎞
⎠
which translates to the equations a = 0, b = 0, c = 0. If you remove that extraneous final column of zeros, you get ⎛
⎝1 0 0 0 1 0 0 0 1
⎞
⎠
This form, a matrix with 1’s on the diagonal and 0 otherwise, is an example inR3 of the result you get when you reduce a matrix of nlinearly independent vectors in Rn. This matrix is called theidentity matrix, with the shorthand I, for reasons that will become clear in the next chapter.
The statement about linearly independent vectors made in the Facts and Notation above can be expressed as the following theorem.
Theorem 3.3
A set ofn vectors A1, . . . , An in Rn is linearly independent if and only if
the echelon form of the matrix whose columns are A1 through An is the ←−
This theorem will grow into a really big theorem over the coming chapters.
identity matrixI.
Exercises
1. Solve each problem by writing an augmented matrix and reducing it to echelon form.
a. Show that (3,1,0), (2,4,3), and (0,−10,−9) are linearly de- pendent.
b. Are (2,1,4), (3,0,1), (7,1,2), and (8,−1,0) linearly depen- dent?
c. Are (4,1,2), (3,0,1), and (7,1,4) linearly dependent?
d. Write (9,8,17) as a linear combination of (2,1,3), (4,1,2), and (7,5,6).
e. Show that (9,8,17), (2,1,3), (4,1,2), and (7,5,6) are linearly dependent.
f. Show that (10,5,10,6) is a linear combination of (4,1,2,0) and (3,2,4,3).
g. Show that (10,5,10,6), (4,1,2,0), and (3,2,4,3) are linearly dependent.
2. a. Show that (6,9,12) is a linear combination of the vectors (1,2,3), (4,5,6), and (7,8,9).
b. Show that (6,9,10) isnot a linear combination of the vectors (1,2,3), (4,5,6), and (7,8,9).
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3. a. Describe all vectors that are orthogonal to (1,0,3) and (7,1,2).
b. Describe all vectors that are orthogonal to (1,0,3), (7,1,2), and (9,1,9).
c. Describe all vectors that are orthogonal to (1,0,3), (7,1,2), and (9,1,8).
d. Which of the three sets of vectors (from parts a, b, and c above) are linearly independent?
e. Compare your answers to parts a, b, and c. How did your answer to part drelate to those answers?
4. Find the intersection of the graphs of X ã(2,1,4) = 8 and X ã
(1,1,5) = 6. Describe that intersection geometrically. ←−
That is, is it a point? a line? a plane? something else?
5. Find the intersection of the graphs of
X = (3,1,2) +t(4,1,6) and X = (−2,5,5) +s(3,−1,1)
6. Write (1,2,5) as a linear combination of (4,1,6) and (3,−1,1).
7. Are the lines with equations X = (3,1,4) +t(0,1,6) and X = Lines are skew in R3 if they are neither parallel nor intersecting. This “third”
possibility can’t exist inR2, right?
(1,1,7) +s(1,2,4) parallel, intersecting, or skew?
8. Show that any set of vectors that containsOis linearly dependent.
9. Complete the second half of the proof of Theorem 3.2 by proving the following statement:
“Given a set of vectors A1, A2, . . . , Ak, if one of the vectors Ai is a linear combination of the other vectors in the set, then the vectors are linearly dependent.”
10. Find numbersa,b, andc, not all zero, so that ←−
Will any other values fora, b, andcwork?
a(7,1,3) +b(2,1,4) +c(8,−1,−6) = (0,0,0)
11. Show that ifa(1,4,7) +b(2,5,8) +c(3,6,0) = (0,0,0), thena=b= c= 0.
12. Find all vectors orthogonal to the rows of
⎛
⎝3 1 2 0 4 1 6 1 1 3 2 0
⎞
⎠
13. Show that any vector orthogonal to the rows of
⎛
⎝1 2 3 4 5 6 7 8 0
⎞
⎠
must be (0,0,0).
3.4 Linear Dependence and Independence
14. When asked to writeD as a linear combination of three vectorsA, B, and C, Sasha ended up with this echelon form:
⎛
⎝1 0 2 0 0 1 −1 0
0 0 0 0
⎞
⎠
a. Is Da linear combination ofA,B, andC? Explain.
b. Are A, B, C, and D linearly dependent? Explain how you know.
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