Linear independence refers to a set of vectors in a vector space that cannot be written as a linear combination of other vectors in the same space. In other words, none of the vectors in the set can be expressed as a multiple of another vector in the set. This concept is important in linear algebra because it allows us to determine the dimension of a vector space.
A set of vectors is linearly independent if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0, where c1, c2, ..., cn are scalars and v1, v2, ..., vn are the vectors in the set, is c1 = c2 = ... = cn = 0. In other words, the only way to get a linear combination of the vectors equal to zero is by setting all the coefficients to zero.
Linear independence refers to a set of vectors that cannot be written as a linear combination of each other, while linear dependence refers to a set of vectors that can be written as a linear combination of each other. In other words, linear independence means that the vectors are "independent" of each other, while linear dependence means that they are "dependent" on each other.
Yes, a set of two vectors can be linearly dependent. For example, if one vector is a multiple of the other, then they are linearly dependent. In this case, one vector can be written as a linear combination of the other by simply multiplying it by the scalar. It is important to note that a set of two vectors can also be linearly independent, as long as they are not multiples of each other.
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 dimension of the vector space is equal to the number of vectors in the set. Additionally, linear independence helps us to solve systems of linear equations and understand the properties of linear transformations.