Linear Algebra: Linear dependence of vectors?

  • #1

Homework Statement


Given:
[tex]v_1 = \left(\begin{array}{cc}1\\-5\\-3\end{array}\right)

v_2 = \left(\begin{array}{cc}-2\\10\\6\end{array}\right)

v_3 = \left(\begin{array}{cc}2\\-9\\h\end{array}\right)

[/tex]

For what value of h is [tex]v_3[/tex] in Span{[tex]v_1, v_2[/tex]} and for what value of h is {[tex]v_1,v_2,v_3[/tex]} linearly dependent?

Homework Equations





The Attempt at a Solution


If my understanding is correct, a vector is in a span of another set of vectors if all vectors are in the same plane? If so then my guess is h would have to be a multiple of -3.

The linear dependence part of the question is getting me though. I was under the impression that all three vectors would have to be multiples of each other for it to be linearly dependent?
 

Answers and Replies

  • #2
For the first question.
How can you express the vector space [itex]V=Span\{u_1,u_2\}[/itex]?
 
  • #3
For the first question.
How can you express the vector space [itex]V=Span\{u_1,u_2\}[/itex]?

A plane?
 
  • #4
Yes, and the equation that desribes that plane, is which?
 
  • #5
HallsofIvy
Science Advisor
Homework Helper
41,833
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Homework Statement


Given:
[tex]v_1 = \left(\begin{array}{cc}1\\-5\\-3\end{array}\right)

v_2 = \left(\begin{array}{cc}-2\\10\\6\end{array}\right)

v_3 = \left(\begin{array}{cc}2\\-9\\h\end{array}\right)

[/tex]

For what value of h is [tex]v_3[/tex] in Span{[tex]v_1, v_2[/tex]} and for what value of h is {[tex]v_1,v_2,v_3[/tex]} linearly dependent?

Homework Equations





The Attempt at a Solution


If my understanding is correct, a vector is in a span of another set of vectors if all vectors are in the same plane?
You understanding is not quite correct. A vector is in a span of another set of vectors if it can be written as a linear combination of the vectors in the set. The span of the set of vectors is a plane only if there are exactly two independent vectors in the set. That is not true in this problem!

If so then my guess is h would have to be a multiple of -3.

The linear dependence part of the question is getting me though. I was under the impression that all three vectors would have to be multiples of each other for it to be linearly dependent?

Are these really the problem you were given? Have you copied the vectors correctly? Notice that -2v1= v2. The span of these two vectors has dimension 1, not 2. v3 will be in their span only if it also is a multiple of either v1 or v2. The first component of v3, 2, is 2 times the first component of v1 (and -1 times the first component of v2) but the second component of v3 is -9 which is not 2 times the second component of v1 (and not -1 times the second component of v2).

v3 while not be in the span of v1 and v2 for any value of h.


(If the second component of v3 were -10, then v3 would be in the span of v1 and v2 for h= -6.)
 
  • #6
The span of the set of vectors is a plane only if there are exactly two independent vectors in the set. That is not true in this problem!
Ahhh, Ok. That makes a lot more sense.

Are these really the problem you were given? Have you copied the vectors correctly?
Yes, these really are the vectors.

The span of these two vectors has dimension 1, not 2.
Just because making assumptions has burned me in the past, you mean the span is a line, correct?

v3 while not be in the span of v1 and v2 for any value of h.
After going back and re-reading the chapter this is what I thought. Thanks for confirming it though.

Since v3 isn't a linear combination of v1 and v2, this means that the set of {v1, v2, v3} has to be linearly independent, right?
 
  • #7
Defennder
Homework Helper
2,591
5
Just because making assumptions has burned me in the past, you mean the span is a line, correct?
Yes, the linear span of v1 and v2 is a line in R3.

Since v3 isn't a linear combination of v1 and v2, this means that the set of {v1, v2, v3} has to be linearly independent, right?
What does it mean for the set to be linearly independent? If it is linearly independent, you should not be able to represent any vector as a linear combination of the other 2. But haven't you just shown (or Halls did) that v1 and v2 are multiples of each other?
 

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