Vector Spaces, Basis Vectors, Subspaces

What is a vector space?

A vector space V is a set which is closed under addition and scalar multiplication. Furthermore, it must have the identity element for addition (zero). Through these properties, we are given many “free” properties of vector spaces, like each element having an inverse.

What is a vector span?

A vector span is the set of all linear combinations of a set of vectors. $span(v_1,...,v_k)=\{c_1{v}_1+...c_k{v}_k\}$ We can think of the span of a matrix as the range of the linear transformation. We consider a set of vectors to be linearly independent if $c_1{v}_1+c_2{v}_2+...+c_k{v}_k=0$ is only true when $c_1=c_2=...=c_k=0$If a matrix has a determinant of zero, then its vectors are linearly dependent.

What is a basis?

A basis of a vector space is a linearly independent set which spans the entire vector space. The dimension of a vector space is given by the number of vectors in its basis. A vector space with a finite basis is finite-dimensional while a vector space with an infinite basis is infinitely dimensional.

What is a subspace?

A subspace of a vector space V is a subset $W\subset V$ that is a vector space in its own right. This means the subspace is closed under addition and multiplication and contains the 0 element. For example, the set of vectors of the form $(x,y,0{)}^T\subset {\mathbb{R}}^3$ is a valid subspace.

What is the theorem of linear dependence and span?

Linearly dependence is only true if and only if there is a non-zero solution to $Ac=0$. So a vector b lies in the span of a matrix A if and only if $Ac=b$ has a solution. This is related to rank. The column rank is the dimension of the span, while row rank is the dimension of the row space of the matrix. LDV Decomposition