10.1 are linear independent ....

In summary, the discussion shows that the functions $e^{2x}$ and $\sin 2x$ are linearly independent on the set of all real numbers, as they cannot be expressed as a linear combination of each other. This concept is important in vector spaces, where vectors must be able to be added and multiplied by scalars.
  • #1
karush
Gold Member
MHB
3,269
5
show that
$e^{2x},\quad \sin 2x$
are linear independent on
$\left\{-\infty,\infty\right\}$

new concept to me
but
$\sin 2x$
has an amplitude
$e^{2x}$
doesn't
 
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  • #2
Whether or not a "vector" has an "amplitude" is not relevant. All we require in a vector space is that we be able to add vectors and multiply vectors by numbers (more generally, "scalars").

Two vectors, u and v, in a vector space are "independent" if the only values or a and b such that au+ bv= 0 are a= b= 0. Here that gives us the equation [tex]ae^{2x}+ bsin(2x)= 0[/tex] for all x. In particular, if x= 0 that becomes [tex]a(1)+ b(0)= a= 0[/tex] and, if [tex]x= \pi/2[/tex], [tex]ae^{\pi/2}+ bsin(\pi/2)= 0[/tex] which, since a= 0, gives [tex]b= 0[/tex].
 
  • #3
SSCwt.png
 
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  • #4
?? That appears to be impossible to read!
 
  • #5
.
 
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FAQ: 10.1 are linear independent ....

What does it mean for a set of vectors to be linearly independent?

Linear independence refers to a set of vectors where none of the vectors can be written as a linear combination of the others. In other words, no vector in the set can be expressed as a sum of multiples of the other vectors.

How do you determine if a set of vectors is linearly independent?

To determine if a set of vectors is linearly independent, you can use the method of Gaussian elimination or the determinant method. In Gaussian elimination, you reduce the matrix formed by the vectors to row echelon form and check for any row of zeros. In the determinant method, you calculate the determinant of the matrix formed by the vectors and check if it is equal to zero.

What is the importance of linear independence in mathematics and science?

Linear independence is important in mathematics and science because it allows us to solve systems of equations, perform transformations, and analyze data. It also plays a crucial role in linear algebra, which is used in many fields such as physics, engineering, and computer science.

Can a set of vectors be linearly independent in one space but linearly dependent in another?

Yes, a set of vectors can be linearly independent in one space but linearly dependent in another. This is because the concept of linear independence is dependent on the dimension of the vector space. A set of vectors may be linearly independent in a higher-dimensional space but become linearly dependent when restricted to a lower-dimensional subspace.

How is linear independence related to the concept of basis?

Linear independence is closely related to the concept of basis. A basis for a vector space is a set of linearly independent vectors that span the entire space. This means that any vector in the space can be written as a linear combination of the basis vectors. Therefore, linear independence is a necessary condition for a set of vectors to form a basis for a vector space.

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