Alpha Floor said:Just take the set of vectors, build a 3x3 matrix out of them and calculate its determinant. If the determinant is not equal to cero, then they are linearly independent. If it is cero, then they are linearly dependent.
EDIT: I have explained it better in this thread (actually the question is almost exactly the same)
ammar555 said:How do I show that for any vectors u,v, and w in a vector space V, the set of the vectors {u-v, v-w, w-u} is a linearly dependent set?
do it in general!
A vector space is a mathematical structure that consists of a set of vectors and operations that can be performed on those vectors. It is a fundamental concept in linear algebra and is used to model various physical and abstract systems.
To show that a set of vectors is in a vector space, you must prove that the set satisfies the 10 axioms or properties of a vector space. These include closure under vector addition and scalar multiplication, existence of a zero vector, and existence of additive and multiplicative inverses for each vector.
Some common examples of vector spaces include the set of all real numbers, the set of all n-dimensional vectors, and the set of all polynomials of degree n or less. Other examples include function spaces, such as the set of all continuous functions or the set of all differentiable functions.
Yes, a vector space can have infinitely many vectors. In fact, most vector spaces have an infinite number of vectors. For example, the set of all real numbers is an infinite vector space.
Vector spaces are used in various scientific fields, such as physics, engineering, and computer science. They provide a powerful mathematical framework for modeling and solving problems involving multidimensional data, physical forces, and transformations. Vector spaces are also essential in quantum mechanics, where they are used to describe the state of a quantum system.