# Linearly independent set in a vector space

• AkilMAI
In summary, if {u;v;w} is a linearly independent set in a vector space, then the set {2u + v + w; u + 2v + w; u + v + 2w} is also linearly independent. This can be proven by creating a system of equations based on the definition of linear independence and showing that the only solution is when all coefficients are equal to 0.

## Homework Statement

I need to prove that, if {u;v;w} is a linearly independent set in a
vector space, then the set
{2u + v + w; u + 2v + w; u + v + 2w}
is also linearly independent.

...

## The Attempt at a Solution

if {u;v;w} is a linearly independent set=> c1*u +c2*v+c3*w=0 and c1=c2=c3=0 and also u,v,w are distinct and different from 0.
=>c1(2u + v + w)+ c2(u + 2v + w)+ c3(u + v + 2w)=0,but if I create a system of equations based on this i get u=v=w=0...any help?

From this upper equation you get a system of equations involving the coefficients c1, c2, c3. This system yields a solution c1 = c2 = c3 = 0. This, in turn, implies what you need to show.

Edit: but note that the c's are not the same as in the upper equation - your equation contains coefficients which are sums of the c's.

writing u(2c1+c2+c3)+v(c1+2c2+c3)+w(c1+c2+2c3)=0...and by definition u v and w have to be distinct and different from 0.=>2c1+c2+c3=0,c1+2c2+c3=0,c1+c2+2c3=0...calculating this system of equations and i get c1=c2=c3=0...is this correct?

AkilMAI said:
writing u(2c1+c2+c3)+v(c1+2c2+c3)+w(c1+c2+2c3)=0...and by definition u v and w have to be distinct and different from 0.=>2c1+c2+c3=0,c1+2c2+c3=0,c1+c2+2c3=0...calculating this system of equations and i get c1=c2=c3=0...is this correct?

Yes. And the important thing is that c1 = c2 = c3 = 0 implies 2c1 + c2 + c3 = 0, c1 + 2c2 + c3 = 0 and c1 + c2 + 2c3 = 0, which is exactly what you need to show.

## What is a linearly independent set in a vector space?

A linearly independent set in a vector space is a group of vectors where no vector can be written as a linear combination of the other vectors. In other words, none of the vectors in the set are redundant or unnecessary to describe the vector space.

## 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 following steps:
1. Write the vectors as columns in a matrix.
2. Use row operations to reduce the matrix to row-echelon form.
3. If all the leading coefficients (first non-zero numbers) are 1, then the vectors are linearly independent. Otherwise, they are linearly dependent.

## Can a set of two vectors be linearly dependent?

Yes, a set of two vectors can be linearly dependent if one vector is a multiple of the other. In other words, if one vector can be written as a scalar multiple of the other, then the set is linearly dependent.

## What is the difference between linearly independent and linearly dependent sets?

The main difference between linearly independent and linearly dependent sets is that in a linearly independent set, none of the vectors can be written as a linear combination of the others. In contrast, in a linearly dependent set, at least one vector can be expressed as a linear combination of the other vectors.

## Why is it important to have linearly independent sets in a vector space?

Linearly independent sets are important in a vector space because they provide a basis for the vector space. This means that any vector in the space can be written as a unique linear combination of the vectors in the set. It also allows for easier calculations and simplification of equations in the vector space.