Two linearly dependent vectors

In summary, the conversation discusses a proof for the statement that two vectors are linearly dependent if and only if one is a scalar multiple of the other. The proof involves considering two cases: when both c1 and c2 are nonzero, and when one of them is zero. The solution also involves separately considering the case where one of the vectors is the zero vector. It is also mentioned that the statement may not hold for a specific linear space V, unless the subspaces are non-trivial.
  • #1
bcjochim07
374
0

Homework Statement



Prove that two vectors are linearly dependent if and only if one is a scalar multiple of the other.

Homework Equations





The Attempt at a Solution

This seems at glance to be a fairly easy proof:

Part I Assume that vectors u and v are linearly dependent.

Then c1u + c2v = 0 where c1 and c2 are not both 0

then u = -c2/c1 * v
and v = -c1/c2 * u But this doesn't make sense to me because what if one of c1 or c2 does equal zero?

Part II Assume that u =av and v=bu , where a and b are constants

then u - av = 0 where the coefficient of u is 1 and v - bu = 0 where the coefficient of v is 1.

Therefore u and v are linearly dependent.

I'm struggling a bit with linear algebra proofs, so any critique or suggestions that anyone could offer would be greatly appreciated.
 
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  • #2
Either c1 or c2 is nonzero. Without loss of generality, let c1 be nonzero. Then since c1u + c2v = 0, dividing by c1, we get u + (c2/c1)v = 0. What does this tell you?
 
  • #3
bcjochim07 said:
Part I Assume that vectors u and v are linearly dependent.

Then c1u + c2v = 0 where c1 and c2 are not both 0

then u = -c2/c1 * v
and v = -c1/c2 * u But this doesn't make sense to me because what if one of c1 or c2 does equal zero?

Look at two different cases:

Case1: c1=0 and c2≠0

Case 2: c2=0 and c1≠0

...your method in partII is not valid, since it neglects the c1=0 case.
 
  • #4
Ok so v = c1u

v -c1u = 0 and suppose that c1 = 0.
v must be the zero vector

now suppose that v-c1=0 and c1 is not equal to zero.
then v-c1u=0 where the coefficients of v and u are both nonzero

Am I going on the right track?

and then I would do a similar thing for u = c2v
 
  • #5
bcjochim07 said:
Ok so v = c1u

v -c1u = 0 and suppose that c1 = 0.
v must be the zero vector

No, start with c1u+c2v=0...as you did before

If c1=0, what can you say about c2? (Remember, the vectors are by assumption not zero vectors)

If c2=0, what can you say about c1?

The case where neither of them are zero was already covered in your 1st post.
 
  • #6
If c1=0, what can you say about c2? : c2 is nonzero

If c2=0, what can you say about c1? c1 is nonzero

so assume c1u + c2v = 0

if c1 is nonzero u = -c2/c1 * v
if c2 is nonzero v= -c1/c2 * u


part II

assume that u and v are scalar multiples of each other:

u = av and v = bu where both a and b are nonzero scalars and u and v are nonzero vectors

u - av = 0 and v - bu = 0 therefore u and v are linearly dependent


the problem also gives a hint that I should separately consider the case where one of the vectors is the zero vector

so assume u = av and v=bu

let u be the zero vector 1u-av = 0 therefore u and v are linearly dependent

let v be the zero vector 1v - bu= 0 therefore u and v are linearly dependent


I don't know if I'm getting this right at all.
 
  • #7
bcjochim07 said:
If c1=0, what can you say about c2? : c2 is nonzero

If c2=0, what can you say about c1? c1 is nonzero

so assume c1u + c2v = 0

if c1 is nonzero u = -c2/c1 * v
if c2 is nonzero v= -c1/c2 * u
Yes, that is correct.


part II

assume that u and v are scalar multiples of each other:

u = av and v = bu where both a and b are nonzero scalars and u and v are nonzero vectors
You are making the same mistake you did before. If u and v are scalar multiples of one another, then u= av and v= bu but it does NOT FOLLOW THAT "both a and b are nonzero scalars and u and v are nonzero vectors". If neither u nor v are zero, then you can say u= av for a non zero and so u- av= 0 with coefficients 1 and -a. If u= 0 then au+ 0v= 0 for any non-zero a and if v= 0, then 0u+ bv= 0 for any non-zero b.

u - av = 0 and v - bu = 0 therefore u and v are linearly dependent


the problem also gives a hint that I should separately consider the case where one of the vectors is the zero vector

so assume u = av and v=bu

let u be the zero vector 1u-av = 0 therefore u and v are linearly dependent

let v be the zero vector 1v - bu= 0 therefore u and v are linearly dependent


I don't know if I'm getting this right at all.
 
  • #8
part i. If c1 is zero and c2 is not zero then for c1u + c2v = 0 to hold, either u = o(vector) or v = 0 and vice versa since it has been said that u and v are linearly dependent
 
  • #9
Is there a linear space V in which the union of any subspaces of V is a subspace except the trivial subspaces V and {0}? pls help
 

1. What does it mean for two vectors to be linearly dependent?

Two vectors are considered linearly dependent if one vector can be written as a scalar multiple of the other vector. This means that one vector can be obtained by multiplying the other vector by a certain number.

2. How can I determine if two vectors are linearly dependent?

To determine if two vectors are linearly dependent, you can use the linear dependence test. This involves setting up a system of equations with the two vectors and solving for the variables. If the system has infinitely many solutions, then the vectors are linearly dependent.

3. What is the geometric significance of linearly dependent vectors?

Geometrically, linearly dependent vectors lie on the same line or are parallel to each other. This can also be seen as one vector being a multiple of the other vector, resulting in them having the same direction.

4. Can more than two vectors be linearly dependent?

Yes, more than two vectors can be linearly dependent. In fact, any set of three or more vectors can be linearly dependent if one vector can be written as a linear combination of the other vectors.

5. What is the significance of linearly dependent vectors in linear algebra?

Linearly dependent vectors play a crucial role in linear algebra as they can help us understand and solve systems of linear equations. They also allow us to simplify calculations and reduce the dimensionality of vector spaces. Additionally, linear dependence can be used to determine if a set of vectors spans a certain vector space.

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