- #1
kiamax
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Homework Statement
Determine whether the vector v1=(!,2,3),v2(3,2,1) and v3(1,1,1) are independent or dependent.
Homework Equations
I'm so lost
The Attempt at a Solution
I'm so lost
It will show whether the vectors are linearly independent or linearly dependent.kiamax said:When I set up the matrix and reduce the rows. This will show if its independent or independent?
Linear independence refers to a set of vectors in a vector space that cannot be written as a linear combination of any other vectors in the same space. In other words, each vector in the set is unique and cannot be expressed as a combination of the others.
A set of vectors is considered linearly independent if the only way to satisfy the equation a1v1 + a2v2 + ... + anvn = 0 is if all of the coefficients a1, a2, ..., an are equal to 0. This means that none of the vectors can be written as a combination of the others.
The number of linearly independent vectors in a vector space is called the dimension of the space. The dimension depends on the number of basis vectors, which are a set of linearly independent vectors that span the space. Therefore, the maximum number of linearly independent vectors a vector space can have is equal to its dimension.
Yes, a set of vectors can be linearly independent in one vector space but not in another. This is because the dimension and basis vectors of each vector space may be different, so a set of vectors that is linearly independent in one space may not be linearly independent in another.
Linear independence is a fundamental concept in linear algebra and mathematics. It plays a crucial role in determining the dimension of a vector space, solving systems of linear equations, and understanding transformations. It is also used in various applications, such as data analysis and machine learning, where linearly independent features are important for accurate predictions.