Linear independence is a concept in linear algebra that describes the relationship between two or more vectors. Vectors are considered linearly independent if none of them can be written as a linear combination of the others. In other words, no vector in the set is redundant or can be expressed as a combination of the others.
To determine if a set of vectors is linearly independent, you can use the linear independence test. This involves setting up a system of equations where each vector is represented by a variable, and then solving for the coefficients. If the only solution is the trivial solution (all coefficients are zero), then the vectors are linearly independent. If there is a non-trivial solution, the vectors are linearly dependent.
No, a set of vectors cannot be both linearly independent and linearly dependent. These are mutually exclusive concepts. If a set of vectors is linearly dependent, it means that at least one vector can be expressed as a linear combination of the others, making it redundant. However, if a set of vectors is linearly independent, no vector can be expressed as a linear combination of the others.
Linear independence is an important concept in linear algebra because it helps us understand the properties and behavior of vectors in a vector space. It is also essential for solving systems of linear equations and finding solutions to problems involving multiple variables. Additionally, linear independence allows us to define a basis for a vector space, which is crucial for representing and manipulating vectors.
The dimension of a vector space is equal to the number of linearly independent vectors that span that space. In other words, the dimension of a vector space is the minimum number of vectors needed to generate all the vectors in that space. This means that the dimension of a vector space is directly related to the linear independence of its vectors.