# Linear Algebra - Linear (in)dependence of a set

• SetepenSeth
In summary, the conversation discusses the linear independence of a set of vectors {u, v, w} on a vector space V. The key question is whether the set {u-v, v-w, u-w} is linearly dependent or independent. The attempt at a solution involves expressing the vectors in terms of their coefficients, forming a matrix, and solving a homogeneous system. However, upon further inspection, it is clear that the third vector is the sum of the first two, making the set linearly dependent. The answer key stating otherwise is incorrect.
SetepenSeth

## Homework Statement

Let { u, v, w} be a set of vectors linearly independent on a vector space V

- Is { u-v, v-w, u-w} linearly dependent or independent?

## Homework Equations

[/B]
A set of vectors u, v, w are linearly independent if for the equation

au + bv + cw= 0 (where a, b, c are real scalars)

The only solution is the trivial one a = b = c = 0

## The Attempt at a Solution

I expressed the vector in base of their coefficients, changed them into column vectors, formed a matrix and solve the homogeneous system

(1)u - (1)v + (0)w
(0)u + (1)v - (1)w
(1)u + (0)v - (1)w

| 1 0 1 |
|-1 1 0 |
| 0 -1 -1 |

After doing (R2 + R1 on R2) > (R3 + R2 on R3) I get:

| 1 0 1|
| 0 1 1|
| 0 0 0|

Which has no pivot on the 3rd column, thus the system has multiple solutions, making the set linearly dependent, however my answer key says is linearly independent and I'm not sure what I may be doing wrong.

Nothing. The key is wrong. If you add the first and second vectors together, you get the third. That set is clearly dependent.

vela said:
Nothing. The key is wrong. If you add the first and second vectors together, you get the third. That set is clearly dependent.

Thank you, I didn't notice that indeed the 3rd vector is the sum of the first two, that is proof enough the key is indeed wrong.

## 1. What is the definition of linear (in)dependence of a set?

Linear (in)dependence of a set refers to the relationship between the vectors within the set. A set of vectors is considered linearly dependent if one or more vectors in the set can be written as a linear combination of the other vectors in the set. On the other hand, a set of vectors is considered linearly independent if none of the vectors can be expressed as a linear combination of the others.

## 2. How do you determine if a set of vectors is linearly dependent?

To determine if a set of vectors is linearly dependent, you can use the row reduction method or the determinant method. If the row reduction method results in a row of zeros, or if the determinant of the matrix formed by the vectors is equal to zero, then the set of vectors is linearly dependent.

## 3. What does it mean for a set of vectors to be linearly independent?

A set of vectors is considered linearly independent if none of the vectors in the set can be expressed as a linear combination of the others. This means that each vector in the set is unique and cannot be duplicated or created by any combination of the other vectors.

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

Yes, a set of two vectors can be linearly dependent. This means that one of the vectors can be written as a linear combination of the other vector. For example, if one vector is a multiple of the other, or if they are parallel, then the set is linearly dependent.

## 5. Why is the concept of linear (in)dependence important in linear algebra?

The concept of linear (in)dependence is important in linear algebra because it helps us understand the relationships between vectors and how they can be combined to create other vectors. It also allows us to solve systems of equations and perform other operations in a more efficient and organized manner.

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