Linear independence is important in linear algebra because it allows us to uniquely represent vectors in a vector space. This means that each vector is necessary and contributes to the span of the vector space. Additionally, linear independence helps us to understand the relationships between vectors and their operations.
A set of vectors is considered linearly independent if no vector in the set can be written as a linear combination of the other vectors. This means that the only solution to the equation c₁v₁ + c₂v₂ + ... + cₙvₙ = 0 is c₁ = c₂ = ... = cₙ = 0, where cᵢ represents a scalar and vᵢ represents a vector in the set.
Linear independence means that a set of vectors cannot be written as a linear combination of the other vectors in the set. Linear dependence, on the other hand, means that a vector can be written as a linear combination of the other vectors in the set.
Yes, a set of three or more vectors can be linearly independent. In fact, the maximum number of linearly independent vectors in a vector space of dimension n is n. So, in a three-dimensional vector space, a set of three linearly independent vectors is possible.
Geometrically, linear independence means that the vectors in a set are not collinear (lying on the same line) or coplanar (lying on the same plane). This means that each vector contributes a unique direction to the span of the vector space.