# Linearly Independent Vectors and Their Span: The Truth Behind <u,v,w> and <u,v>

• macca1994
In summary, the conversation is discussing whether <u,v,w> is equal to <u,v> if u,v,w are linearly independent. The person asking the question is unsure but suggests that it might be true since the span <u,v> would not include any multiple of w. The other person provides a proof by contradiction, showing that assuming <u,v,w> is equal to <u,v> would result in a non-trivial solution that contradicts the definition of linear independence. The conversation ends with confirmation that this is the correct reasoning.
macca1994

## Homework Statement

Suppose u,v,w are linearly independent, is it true that <u,v,w> does not equal <u,v>

## The Attempt at a Solution

I started by defining what it meant to be linearly independent but am unsure where to go from there. I think the statement is true since the span <u,v> won't include any multiple of w but i can't give a solid proof

Assume by contradiction that <u,v,w>=<u,v>. Then $w\in <u,v>$. Thus...

oh i think i get it, by assuming that we can say by definition

w=λ1u + λ2v
then there is now a non trivial solution to
λ1u + λ2v + λ3w=0 which contradicts the statement that u,v,w are linearly independent, is that right?

That's right!

Sweet, cheers

## 1. What is the definition of "linearly independent vectors"?

Linearly independent vectors are a set of vectors where no vector in the set can be represented as a linear combination of the other vectors in the set. In other words, there is no non-trivial solution to the equation a + b + c = 0, where a, b, and c are scalars and , , and are vectors.

## 2. How can I determine if a set of vectors is linearly independent?

A set of vectors is linearly independent if and only if the determinant of the matrix formed by the vectors is non-zero. This can be determined by using Gaussian elimination or calculating the determinant directly.

## 3. What is the importance of linearly independent vectors?

Linearly independent vectors are important because they form a basis for a vector space. This means that any vector in the space can be represented as a unique linear combination of the basis vectors. This also allows for easier computation and manipulation of vectors within the space.

## 4. Can a set of more than three vectors be linearly independent?

Yes, a set of any number of vectors can be linearly independent as long as none of the vectors can be represented as a linear combination of the others. However, in a three-dimensional space, the maximum number of linearly independent vectors is three.

## 5. What is the relationship between linearly independent vectors and vector span?

The span of a set of vectors is the set of all possible linear combinations of those vectors. If the vectors are linearly independent, then their span is the entire vector space. However, if the vectors are linearly dependent, their span is a subspace of the vector space.

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