Question

Someone explain and show how finding a subspace works and knowing how it is one with a matrix example.

Answer 1

A subspace is a vector space that is a subset of some other (higher-dimension) vector space. If that S and R are two vector spaces and that S is a subset of R. Then S is a subspace of R if:

1. The zero vector, 0, is in S.
2. If u and v are elements of S, then the sum u + v is an element of S.
3. If u is in S and c is a scalar, then cu (the scalar product) is in S

Subspaces associated with matrices

Let A be an m × n matrix.

1. The row space of A is the subspace row(A) of R n spanned by the rows of A

2. The column space of A is the subspace col(A) of R m spanned by the columns of A

3. The null space of A is the subspace of R consisting of solutions such that Ax = 0. It is denoted by null(A)

Let us now find a basis for subspace V = {x= R^3 I Ax = 0} where any matrix say A = 2 0 5

1 1 -1/2

Basis for subspace V implies a set of vectors in V such that they span V and are linearly independent

Now,we have to solve for V such that Ax = 0 as this the relationship that defines this V to the matrix A

CALCULATING THE NULL SPACE

Hence we have calculated the basis for V such that Ax = 0