# Use identities to Prove linearly idenpendent

In summary, identities in linear algebra are mathematical expressions that are always true and can be used to simplify and prove equations and expressions. They can also be used to prove linear independence by showing that a linear combination of vectors is equal to zero or not. Identities can be used to prove linear independence of any set of vectors, but it is important to choose the appropriate ones for the specific set. Some commonly used identities for this purpose include the Zero Vector Identity, the Distributive Property Identity, and the Associative Property Identity. In real-world applications, identities can be used to prove the independence of variables or parameters, which is important in fields such as economics, engineering, and physics for analyzing systems and making predictions.
determine following setd of vectors in F(-infinitity, infinitiy) are linearly independent (using appropriate identities)

0, [cos (pi*x)]^3 , [sin 3*pi*x]^5

pls help !

One of your vectors is 0? A set containing the 0 vector cannot be independent!

that mean any set of vectors contain a 0 cannot be linearly independent ?
not independent but linealy independent

That's what he meant.

## 1. What are identities in linear algebra?

Identities in linear algebra refer to mathematical expressions that are always true, regardless of the values of the variables involved. They are used to simplify and prove equations and expressions.

## 2. How do identities help in proving linear independence?

Identities can be used to show that a linear combination of vectors is equal to zero. If this is true, then the vectors are linearly dependent. Conversely, if the linear combination is not equal to zero, then the vectors are linearly independent.

## 3. Can identities be used to prove linear independence of any set of vectors?

Yes, identities can be used to prove linear independence of any set of vectors. However, it is important to choose the appropriate identities for the specific set of vectors being analyzed.

## 4. Are there any specific identities that are commonly used to prove linear independence?

Yes, there are several commonly used identities in linear algebra that are used to prove linear independence. These include the Zero Vector Identity, the Distributive Property Identity, and the Associative Property Identity.

## 5. How can identities be used to prove linear independence in real-world applications?

In real-world applications, identities can be used to prove that a set of variables or parameters are independent of each other. This can be helpful in various fields such as economics, engineering, and physics, where linear independence is important in analyzing systems and making predictions.

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