# Linear algebra, field morphisms and linear independence

• mariang
In summary, we are asked to prove that a set of field morphisms, f1, f2, ..., fn, are linearly independent over a field K, given that fi != fj when i != j and Ker(fi) = 0 for all i.
mariang

## Homework Statement

Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K.

## Homework Equations

f1, ..., fn are field morphisms => Ker (fi) = 0 (injective)

## The Attempt at a Solution

I tried to use the linearity and the injectivity but i got stuck.

mariang said:

## Homework Statement

Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K.

## Homework Equations

f1, ..., fn are field morphisms => Ker (fi) = 0 (injective)

## The Attempt at a Solution

I tried to use the linearity and the injectivity but i got stuck.
In such cases it is often helpful to start with ##n=1## and ##n=2## to see how possible arguments work. From there one can either proceed by a general ##n## or per induction. I assume we also have to require ##f_i \neq 0\,.##

## 1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of vector spaces and linear transformations between them. It involves the manipulation of linear equations and their solutions, as well as the study of geometric concepts such as lines, planes, and higher-dimensional objects.

## 2. What are field morphisms?

Field morphisms, also known as homomorphisms, are functions that preserve the algebraic structure of a field. In other words, they map elements from one field to another in a way that preserves the operations of addition, subtraction, multiplication, and division.

## 3. Why is linear independence important?

Linear independence is important because it allows us to represent a set of vectors in a concise and efficient manner. It also enables us to perform operations such as matrix inversion and solving systems of linear equations, which are essential in many areas of mathematics and science.

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

A set of vectors is linearly independent if none of the vectors in the set can be written as a linear combination of the other vectors. In other words, no vector in the set is redundant and each vector adds new information to the set. This can be determined by setting up a system of linear equations and solving for the coefficients, or by using other methods such as the determinant or rank of a matrix.

## 5. Can you give an example of a real-world application of linear algebra?

Linear algebra has many real-world applications, such as in computer graphics, data analysis, and machine learning. One example is in image processing, where linear algebra is used to manipulate and enhance images through techniques such as rotation, scaling, and blurring. It is also used in economics and finance to model and analyze systems of equations, such as supply and demand curves or stock market trends.

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