You assume that v is nonzero - that's what "if v is nonzero" means. You don't need to show the things that you are assuming.ashina14 said:How can I show v is non zero?
A vector space is a mathematical structure consisting of a set of objects called vectors, which can be added together and multiplied by numbers, called scalars, to produce new vectors. It is a fundamental concept in linear algebra and is used to study geometric properties and transformations.
A vector space must satisfy several properties, including closure under vector addition and scalar multiplication, associativity and commutativity of addition, and the existence of a zero vector and additive inverses for each vector. It must also follow the distributive property of scalar multiplication over vector addition.
A vector space is a more general concept than a Euclidean space. While a Euclidean space is a vector space, not all vector spaces are Euclidean spaces. A Euclidean space has additional structure, such as a defined distance and angle between vectors, while a vector space only has the operations of vector addition and scalar multiplication.
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 with real coefficients. Other examples include the set of all continuous functions on a given interval and the set of all square matrices of a given size.
The dimension of a vector space is the number of vectors in a basis for that space. A basis is a set of linearly independent vectors that span the entire space. The dimension of a vector space can also be thought of as the number of coordinates needed to specify any vector in that space.