LinAlg - Due today (Linear independence)

In summary, the given vectors A, B, and C are determined to be linearly dependent. To find a non-trivial linear relation, solve the equation aA+bB+cC=0, where a, b, and c are coefficients. The first equation obtained is -2a-2b=0, and by substituting a=-b into the remaining equations, multiple solutions can be obtained.
  • #1
eiktmywib
3
0

Homework Statement


A=[-2
-7
-1]

B=[-2
-4
-3]

C=[0
6
-4]

Determine whether or not the three vectors listed above are linearly independent or linearly dependent.
I have determined that they are linearly DEPENDENT.

If they are linearly dependent, determine a non-trivial linear relation - (a non-trivial relation is three numbers which are not all three zero.) otherwise, if the vectors are linearly independent, enter 0's for the coefficients, since that relationship always holds.


Homework Equations


none, sorry

The Attempt at a Solution


I have put the vectors into a matrix and solved it... but then I got something strange, and everytime I do it... I always get different answers. I have determined that they are linearly dependent though.
 
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  • #2
please help... I have no idea what to do =(
 
  • #3
You want to solve aA+bB+cC=0. If you equate the components you get three equations in the three unknowns. The first one is -2a-2b=0. What are the others? If you know they are linearly dependent you should expect to get lots of solutions. Your first one says a=-b. I would substitute that into the other equations.
 

1. What is linear independence in Linear Algebra?

Linear independence refers to a set of vectors in a vector space that cannot be written as a linear combination of other vectors in that space. In other words, no vector in the set is redundant or can be expressed as a combination of other vectors in the set. This concept is important in determining the dimension of a vector space and solving systems of linear equations.

2. How is linear independence determined?

Linear independence is determined by setting up a system of equations using the vectors in question. If the only solution to the system is the trivial solution (all variables equal to 0), then the vectors are linearly independent. If there are other non-trivial solutions, then the vectors are linearly dependent.

3. What is the significance of linear independence?

Linear independence is important in many areas of mathematics and science, including linear algebra, differential equations, and physics. It allows us to determine the dimension of a vector space, solve systems of linear equations, and understand the relationships between vectors within a space.

4. Can a set of vectors be both linearly independent and linearly dependent?

No, a set of vectors cannot be both linearly independent and linearly dependent. This is because if a set of vectors is linearly independent, it means that no vector in the set is a linear combination of other vectors in the set. If a set of vectors is linearly dependent, it means that at least one vector in the set can be expressed as a linear combination of other vectors in the set. These two concepts are mutually exclusive.

5. How is linear independence used in real-world applications?

Linear independence is used in a variety of real-world applications, including computer graphics, data analysis, and engineering. It allows us to understand the relationships between variables and make predictions based on those relationships. For example, in computer graphics, linear independence is used to rotate and scale objects in a 3D space. In data analysis, it is used to identify important variables and reduce the dimensionality of data sets. In engineering, it is used to solve systems of equations and model physical systems.

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