- #1
philipc
- 57
- 0
I was just wondering if I can make this statement, if I was to prove any three vectors of R^3 are linear independent, can I also say those three vectors span R^3?
Philip
Philip
Linear independence refers to the relationship between vectors in a vector space. In R^3, it means that the three vectors are not multiples of each other and cannot be written as a linear combination of each other. This means that none of the vectors can be expressed as a sum or difference of the other two vectors multiplied by a scalar.
To prove linear independence, we need to show that the only solution to the equation a1v1 + a2v2 + a3v3 = 0 (where a1, a2, and a3 are scalars and v1, v2, and v3 are the three vectors) is a1 = a2 = a3 = 0. This can be done by setting up a system of equations and solving for the scalars.
Proving linear independence of vectors is important because it allows us to determine if a set of vectors can form a basis for a vector space. In other words, if a set of vectors is linearly independent, then we can use them to span the entire vector space and any vector in that space can be written as a linear combination of those vectors.
If three vectors in R^3 are not linearly independent, then it means that at least one of the vectors can be expressed as a linear combination of the other two. This can lead to redundancy and inefficiency in operations involving those vectors.
There are a few shortcut methods that can be used to prove linear independence, such as using determinants or checking the rank of a matrix formed by the vectors. However, these methods may not always be applicable and the most reliable way to prove linear independence is by using the definition and solving for the scalars as mentioned in question 2.