Let us study the Nullspace of a Matrix,
The solution sets of homogeneous linear systems provide an important source of vector spaces. Let A be an m by n matrix, and consider the homogeneous system
Since A is m by n, the set of all vectors x which satisfy this equation forms a subset of R n . (This subset is nonempty, since it clearly contains the zero vector: x = 0 always satisfies A x = 0.) This subset actually forms a subspace of R n , called the nullspace of the matrix A and denoted N(A). To prove that N(A) is a subspace of R n , closure under both addition and scalar multiplication must be established. If x1 and x2 are in N(A), then, by definition, A x1 = 0 and A x2 = 0. Adding these equations yields
which verifies closure under addition. Next, if x is in N(A), then A x = 0, so if k is any scalar,
verifying closure under scalar multiplication. Thus, the solution set of a homogeneous linear system forms a vector space. Note carefully that if the system is not homogeneous, then the set of solutions is not a vector space since the set will not contain the zero vector.
Hope the above explanation helped you to know about Nullspace of a matrix, now let me explain you about matrices and determinants.
The solution sets of homogeneous linear systems provide an important source of vector spaces. Let A be an m by n matrix, and consider the homogeneous system
Ax = 0
Since A is m by n, the set of all vectors x which satisfy this equation forms a subset of R n . (This subset is nonempty, since it clearly contains the zero vector: x = 0 always satisfies A x = 0.) This subset actually forms a subspace of R n , called the nullspace of the matrix A and denoted N(A). To prove that N(A) is a subspace of R n , closure under both addition and scalar multiplication must be established. If x1 and x2 are in N(A), then, by definition, A x1 = 0 and A x2 = 0. Adding these equations yields
which verifies closure under addition. Next, if x is in N(A), then A x = 0, so if k is any scalar,
verifying closure under scalar multiplication. Thus, the solution set of a homogeneous linear system forms a vector space. Note carefully that if the system is not homogeneous, then the set of solutions is not a vector space since the set will not contain the zero vector.Hope the above explanation helped you to know about Nullspace of a matrix, now let me explain you about matrices and determinants.
