# Answer to Question #24298 in Linear Algebra for Matthew Lind

Question #24298

Let A be an N by N matrix with rank J<N. Prove then that there are N - J linearly independent solutions of the system Ax = 0, and the null space of A has dimension N - J.

Expert's answer

Every solution is the solutionto Bx=0, where B is row echelon form. For B we have n-j last zero rows, and

these corresponding x_i 's can be choosen linearly independent, and other will

be automatically computed, so there will be n-j lin indep solutions.

these corresponding x_i 's can be choosen linearly independent, and other will

be automatically computed, so there will be n-j lin indep solutions.

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