For which L(s) will be these vectors linearly dependent?

In summary, the conversation is about determining the value of L that would make three given vectors (a, b, c) linearly dependent. The speaker provides three equations and finds that L can be either 1 or 2, but is unsure if there are any other possible values. They also mention a second question about the dependence of ß on L if a new vector v is in the span of a, b, and c. The speaker clarifies that L is one of the given values and asks for confirmation of the definition of linearly dependent.
  • #1
gotmejerry
9
0
So i have 3 vectors:
a= [1 1 1]
b= [2 L 0]
c= [L 2 3]

How do I calculate the L in order to make these vecotrs linearly dependent?

How does ß depend from L if v= [ß 0 -1] and v is in span(a b c)?

Thank you!
 
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  • #2
Do you know what "linearly dependent" means? Use the definition of "linearly dependent" You will get a cubic equation for L but, fortunately, there's 1 obvious root.

For the second question, are we to assume that L is one of those values? Otherwise, the span of a, b, and c is all of R3 and [itex]\beta[/itex] can be anything.
 
  • #3
I guess I know.

I wrote 3 equations:

α + 2β + Lγ = 0
α + Lβ + 2γ = 0
α + + 3γ = 0

And i got, L can be 1 or 2. Then I checked it and for these Ls the vectors are dependents. But how do I know that there aren't more Ls.

For the second question. Yes L is from those values.
 
  • #4
HallsofIvy said:
Do you know what "linearly dependent" means? Use the definition of "linearly dependent" You will get a cubic equation for L but, fortunately, there's 1 obvious root.

For the second question, are we to assume that L is one of those values? Otherwise, the span of a, b, and c is all of R3 and [itex]\beta[/itex] can be anything.

But I maybe got the definition wrong.
 
  • #5


To determine the values of L that will make these vectors linearly dependent, we can use the fact that linearly dependent vectors can be written as a linear combination of each other. This means that one vector can be expressed as a multiple of another vector, or a linear combination of two other vectors.

In this case, we can start by setting up an equation using the three given vectors:

x * a + y * b + z * c = 0

Where x, y, and z are constants. We can then substitute the values of a, b, and c into this equation and solve for L:

x * [1 1 1] + y * [2 L 0] + z * [L 2 3] = 0

Expanding this equation, we get:

[x + 2y + z, x + y + 2z, x + y + 3z] = 0

Since we know that linearly dependent vectors must have a non-trivial solution (where not all constants are equal to zero), we can set one of the constants to be equal to 1 and solve for the others. Let's set x=1:

[1 + 2y + z, 1 + y + 2z, 1 + y + 3z] = 0

From this equation, we can see that y = -1 and z = 1. Substituting these values back into the original equation, we get:

1 * [1 1 1] + (-1) * [2 L 0] + 1 * [L 2 3] = 0

This means that for these three vectors to be linearly dependent, L must equal -2.

For the second question, if v = [ß 0 -1] is in the span of the three given vectors, then it can be written as a linear combination of them. This means that there exist constants x, y, and z such that:

x * [1 1 1] + y * [2 L 0] + z * [L 2 3] = [ß 0 -1]

We can use the same method as before to solve for the values of x, y, and z in terms of L. This will give us a relationship between ß and L, showing how they are dependent on each other in order for v
 

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

Linear dependence refers to a situation where one or more vectors in a set can be written as a linear combination of the other vectors in the set. In other words, one or more vectors in the set can be expressed as a scalar multiple of another vector in the same set.

2. How can I determine if a set of vectors is linearly dependent?

One way to determine if a set of vectors is linearly dependent is by using the determinant method. If the determinant of the matrix formed by the vectors is equal to 0, then the vectors are linearly dependent. Another method is to try and write one vector as a linear combination of the others. If this is possible, then the vectors are linearly dependent.

3. Can a set of only two vectors be linearly dependent?

Yes, a set of only two vectors can be linearly dependent. This can happen when one vector is a scalar multiple of the other vector. In this case, one vector can be written as a linear combination of the other, making the set linearly dependent.

4. What is the difference between linear independence and linear dependence?

Linear independence refers to a situation where none of the vectors in a set can be written as a linear combination of the other vectors in the set. In other words, each vector in the set is unique and cannot be expressed as a scalar multiple of another vector. Linear dependence, on the other hand, occurs when one or more vectors in a set can be written as a linear combination of the other vectors in the set.

5. Is it possible for a set of vectors to be both linearly independent and linearly dependent?

No, it is not possible for a set of vectors to be both linearly independent and linearly dependent. These two concepts are mutually exclusive. A set can either be linearly independent or linearly dependent, but not both at the same time.

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