The rank-nullity theorem states that rank of A + nullity of A = number of columns of A.
Think of the theorem it this way: in the row reduced echelon form of A: 1) the number of pivot columns is the rank of A, 2) the number of non-pivot columns is the number of free variables in the solution of Ax = 0 (which is the nullity of A), and 3) the number of pivot columns plus the the number of non-pivot columns is the number of columns (note that the number of columns in the row reduced echelon form of A is the same as the number of columns in A).
In this problem, 4 + nullity of A = 7, so nullity of A = 3.
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The rank-nullity theorem states that rank of A + nullity of A = number of columns of A.
Think of the theorem it this way: in the row reduced echelon form of A: 1) the number of pivot columns is the rank of A, 2) the number of non-pivot columns is the number of free variables in the solution of Ax = 0 (which is the nullity of A), and 3) the number of pivot columns plus the the number of non-pivot columns is the number of columns (note that the number of columns in the row reduced echelon form of A is the same as the number of columns in A).
In this problem, 4 + nullity of A = 7, so nullity of A = 3.
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