# Prove Au and Av Linearly Independent

In summary, linear independence refers to two vectors that cannot be written as a multiple of each other, and can be proven by setting up a system of equations where the only solution is when the coefficients are both 0. This method can be used for more than two vectors, which can span a higher-dimensional vector space. Geometrically, linear independence means that the vectors do not lie on the same line and are not redundant in expressing a given vector space.

## Homework Statement

Let u_1...u_n be linearly independent column vectors in R^n and A an invertible n x n matrix. Prove that the vectors Au_1...Au_n are linearly independent.

## The Attempt at a Solution

It is easy to prove this using scalars and the definition of linear independence. But, then why is this relevant to invertible matrices? Is there a way to prove this using column spaces, row spaces, null spaces, etc?

It is not true if A is not invertible!

## 1. What does it mean for two vectors to be linearly independent?

Two vectors are considered linearly independent if there is no way to write one of the vectors as a scalar multiple of the other. In other words, they do not lie on the same line and are not redundant in expressing a given vector space.

## 2. How can I prove that two vectors are linearly independent?

To prove that two vectors, Au and Av, are linearly independent, you can use the definition of linear independence. This means that you need to show that the only solution to the equation cAu + dAv = 0 (where c and d are scalars) is when c = d = 0. You can do this by setting up a system of equations and solving for c and d.

## 3. Can you use a system of equations to prove linear independence?

Yes, using a system of equations is a common method for proving linear independence. This method involves setting up a system of equations with the two vectors and their corresponding coefficients. If the only solution to this system is c = d = 0, then the vectors are linearly independent.

## 4. What is the geometric interpretation of linear independence?

Geometrically, linear independence means that the two vectors do not lie on the same line. This can also be interpreted as the vectors pointing in different directions and not being redundant in expressing a given vector space.

## 5. Can you have more than two linearly independent vectors?

Yes, it is possible to have more than two linearly independent vectors. In fact, the number of linearly independent vectors in a given vector space is known as the dimension of that space. This means that a higher number of linearly independent vectors can span a higher-dimensional vector space.

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