By Prabhat Choudhary

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The dimension dim V of a vector space V is the number of vectors in a basis. 37 Systems of Linear Equations For a vector space consisting only of zero vector 0 we put dim V = o. If V does not have a (finite) basis, we put dim V = 00. If dim V is finite, we call the space V finitedimensional; otherwise we call it infinite-dimensional. , that it does not"depend on the choice of a basis. This immediately implies the following Proposition. A vector space Vis finite-dimensional if and only if it has a finite spanning system.

To see that vectors w I ,w2' ... , wr span the row space, one can notice that row operations do not change the row space. This can be obtained directly from analyzing row operations, but we present here a more formal way to demonstrate this fact. 44 Systems of Linear Equations For a transformation A and a set X let us denote by A(X) the set of all elements y which can represented as y = A(x), x E X, A(X) : = {y = A(x) : x EX}. If a is an m x n matrix, and Ae is its echelon form, Ae is obtained from A be left multiplication Ae = EA, where E is an m x m invertible matrix (the product of the corresponding elementary matrices).

Suppose that A is an m x n matrix, B is an n x p matrix, and that c E JR. Then c(AB) = (cA)B = A(cB). Systems of Linear Equations Note that the system of linear equations can be written in matrix form as Ax = b, where the matrices A, x and b are given. We shall establish the following important result. Proposition. Every system of linear equations of the form,has either no solution, one solution or infinitely many solutions. Proof Clearly the system (2) has either no solution, exactly one solution, or more than one solution.