Please give an example of Vector span and subspaces and a b

Please give an example of ( Vector span and subspaces) and a brief explanation of what they are and how to find them

Solution

Let us consider the vector space R² . Any vector (x, y)^T ( where x, y are arbitrary real valued scalars) can be expressed as x(1,0)^T +y(0,1)^T i.e. as a linear combination of the vectors (1,0)^T and (0,1)^T. Hence R² = pan { (1,0)^T, (0,1)^T} . The span of a set of vectors means the set of all linear combinations of the vectors in the set. Now, let us check whether (x+iy, a+ib)^T is in span { (1,0)^T, (0,1)^T}. ( here i = -1). Apparently, since x+iy and a+ib are not real valued scalars ( unless y = 0 =b), hence the vector (x+iy, a+ib)^T is not in span { (1,0)^T, (0,1)^T} i.e. R². Similarly, the vector (x,y,z)^T is not in span { (1,0)^T, (0,1)^T} i.e. R² since (x,y,z)^T cannot be expressed as a linear combination of the vectors (1,0)^T and (0,1)^T.

Now, let us consider V = span { (x, y)^T : x, y R and x –y = 0}. Then V = span{(x, x)^T : x R } i.e. V is the line y = x in the xy plane. Clearly V is a subspace of R² as V R² and it satisfies all the axioms of a vector space. A subspace of a vector space W is a subset of W , which is a vector space by itself.