Linear independence refers to a set of vectors where none of the vectors can be written as a linear combination of the others. In other words, no vector in the set can be expressed as a sum of multiples of the other vectors.
To determine if a set of vectors is linearly independent, you can use the method of Gaussian elimination or the determinant method. In Gaussian elimination, you reduce the matrix formed by the vectors to row echelon form and check for any row of zeros. In the determinant method, you calculate the determinant of the matrix formed by the vectors and check if it is equal to zero.
Linear independence is important in mathematics and science because it allows us to solve systems of equations, perform transformations, and analyze data. It also plays a crucial role in linear algebra, which is used in many fields such as physics, engineering, and computer science.
Yes, a set of vectors can be linearly independent in one space but linearly dependent in another. This is because the concept of linear independence is dependent on the dimension of the vector space. A set of vectors may be linearly independent in a higher-dimensional space but become linearly dependent when restricted to a lower-dimensional subspace.
Linear independence is closely related to the concept of basis. A basis for a vector space is a set of linearly independent vectors that span the entire space. This means that any vector in the space can be written as a linear combination of the basis vectors. Therefore, linear independence is a necessary condition for a set of vectors to form a basis for a vector space.