Linearly independent set in a vector space

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.
  • #1
77
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.



Homework Equations


...


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?
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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?
 
  • #4
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.

Suggested for: Linearly independent set in a vector space

Back
Top