ANSWER THIS QUESTION BRIEFLY IN ONE TO TWO SENTENCES What ne

ANSWER THIS QUESTION BRIEFLY IN ONE TO TWO SENTENCES!

What new information have we learned about how the four fundamental subspaces are related?

Solution

The four fundamental subspaces of a matrix A are the column space, null space, row space, and left null space ( null space of A^T ). The bases of all four spaces can be obtained using Gaussian elimination, and some of them are orthogonal to each other (Row Space and Null Space of a matrix are orthogonal to each other). (Same for other two) There are close relationships between the dimensions of all four spaces, and the sum of the dimensions of the row and column spaces equals the rank of A.   

The kernel of the matrix A (null space) is the set of all x such that Ax=0 is important for understanding the solutions to several matrix equations. If x₀   is a solution to Ax=b, then every other solution is given by x0 +k where k is in the null space.

The null space of A^T is just another way of describing the \"left null space\" of A, since xA=0 iff AT x^T =0 . The null space is now the set of all x such that xA=0, and we can draw the same conclusions about solutions to xA=b.In short, these four spaces (really just two spaces, with a left and a right version of the pair) carry all the information about the image and kernel of the linear transformation that A is affecting, whether we are using it on the right or on the left.