Linear dependence refers to the relationship between two or more vectors in a vector space. If one vector can be expressed as a linear combination of the other vectors, then the vectors are considered linearly dependent. In other words, one vector can be written as a multiple of another vector, making it redundant in the set.
To determine if a set of vectors is linearly dependent or independent, you can use the linear dependence test. This test involves setting up a system of equations using the vectors and solving for the coefficients. If there is a non-trivial solution (where at least one coefficient is not equal to zero), then the vectors are linearly dependent. If the only solution is when all coefficients are equal to zero, then the vectors are linearly independent.
An example of linearly dependent vectors in real life would be a set of three vectors representing the forces acting on an object in three dimensions. If one of the vectors can be written as a combination of the other two, then they are linearly dependent. An example of linearly independent vectors could be the three primary colors in an RGB color model, as they cannot be expressed as a combination of the other two colors.
Linear independence is important in linear algebra because it allows us to determine the dimension of a vector space. If a set of vectors is linearly independent, then the number of vectors in the set is equal to the dimension of the vector space. This concept is also crucial in solving systems of linear equations and determining the uniqueness of solutions.
The rank of a matrix is equal to the number of linearly independent columns or rows. This means that the rank is a measure of the linear independence of the vectors in a matrix. If the rank is equal to the number of columns or rows, then the vectors are linearly independent. If the rank is less than the number of columns or rows, then the vectors are linearly dependent.