Linear (in)dependent vector sets

In summary, linearly dependent vector sets are a collection of vectors in a vector space where at least one vector can be expressed as a linear combination of the other vectors. To determine if a set of vectors is linearly dependent, the linear dependence test can be used. Understanding linearly dependent vector sets is important for determining if a set of vectors can span a vector space and simplifying calculations. A set of only two vectors can be linearly dependent, and they are commonly used in real-world applications such as computer graphics, engineering, and physics.
  • #1
madking153
37
0
are 6, 3*(sinx)^2 and 2*(cosx)^2 - linearly independent ?

Can someone explain how to determine it ?
 
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  • #2
What have you tried?

You might want to make use of the Pythagorean identity (that sin^2(x) + cos^2(x) = 1).
 
  • #3
how i use the identity to solve that problem
 
  • #4
well, what does it mean for things to be linearly independent?
 

What is the definition of linearly dependent vector sets?

Linearly dependent vector sets are a collection of vectors in a vector space where at least one of the vectors can be expressed as a linear combination of the other vectors.

How can I determine if a set of vectors is linearly dependent or independent?

To determine if a set of vectors is linearly dependent, you can use the linear dependence test. This involves setting up a system of equations with the vectors as variables, and solving for the coefficients. If there are infinitely many solutions, the vectors are linearly dependent. If there is a unique solution, the vectors are linearly independent.

Why is it important to understand linearly dependent vector sets?

Understanding linearly dependent vector sets is important because it helps us to determine if a set of vectors can span a vector space. If a set of vectors is linearly dependent, it means that some of the vectors are redundant and do not contribute to the span of the vector space. This can help us to simplify calculations and solve problems more efficiently.

Can a set of only two vectors be linearly dependent?

Yes, a set of only two vectors can be linearly dependent. For example, if one vector is a multiple of the other, then they are linearly dependent. In general, if a set of vectors has more vectors than the dimension of the vector space, it is highly likely that the set is linearly dependent.

How can I use linearly dependent vector sets in real-world applications?

Linearly dependent vector sets are used in a variety of real-world applications, such as computer graphics, engineering, and physics. For example, in computer graphics, linearly dependent vector sets can be used to represent transformations of objects in a 3D space. In engineering, they can be used to model systems and analyze their behavior. In physics, they can be used to represent forces acting on an object and determine its motion.

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