Question

Do the dimensions of the column space and the null space add up to the number of columns of a matrix?

a) True
b) False

Answer 1

The statement that the dimensions of the column space and the null space add up to the number of columns of a matrix is true, as per the Rank-Nullity Theorem in linear algebra.

Step-by-step explanation:

The correct statement is true: the dimensions of the column space (also known as the rank of the matrix) and the null space (also known as the nullity of the matrix) add up to the number of columns of a matrix. This fact is known as the Rank-Nullity Theorem in linear algebra, which states that for any matrix A with n columns, the rank of A plus the nullity of A is equal to n. Essentially, this theorem provides a fundamental relationship between the linearly independent columns of a matrix (column space) and the solutions to the homogeneous equation Ax = 0 (null space).