# Solution space

ajayguhan
How is the dimension of solution space is n-r, where n is the number of unknowns and r is the rank of A.

Just perform Gauss-Jordan elimination on the matrix and you see it immediately in the resulting matrix.

ajayguhan
Rank is the number of lineraly independent vector of a matrix.
Dimension is the number of lineraly independant vector of vector space.
So rank r should be the dimension of the solution space, isn't it?

But the column vectors of the matrix do not lie in the solution space, do they?

(With "solution space" I assume that you mean what is commonly called the null space of the matrix A, i.e. the set of vectors x, which satisfy Ax=0.)

If the matrix A is of type m x n, which vector spaces Rk are the row space, column space, and null space subspaces of, respectively? And which vector spaces have a dimension equal to the rank of A?

ajayguhan
Solution space must have a dimension equal to rank A.
But it is been stated that dimension of solution space in n-r , so i didn't get it!