Gear300 said:It is stated that for n-dimensional Euclidean space, n vectors are needed at least for linear independence. But if an n-dimensional Euclidean space also includes (n-1)-dimensional Euclidean space, then why can't it also include a family of n-1 linearly independent vectors?
Linear dependence refers to the relationship between two or more vectors in a vector space. A set of vectors is considered linearly dependent if at least one of the vectors in the set can be expressed as a linear combination of the other vectors.
Linear dependence can be determined by using the definition of linear dependence mentioned above. Additionally, a set of vectors is considered linearly dependent if the determinant of the matrix formed by the vectors is equal to 0.
The main difference between linear independence and linear dependence is that in linear independence, none of the vectors in the set can be expressed as a linear combination of the other vectors. In linear dependence, at least one vector can be expressed as a linear combination of the other vectors.
Linear dependence is important in linear algebra because it helps determine whether a set of vectors spans the entire vector space. If a set of vectors is linearly dependent, it means that some of the vectors can be removed without changing the span of the set, making it easier to work with and analyze.
Linear dependence is used in real-world applications, such as data analysis and machine learning, to identify and eliminate redundant or irrelevant features. In economics, linear dependence can also help determine the relationships between different variables in a system.