The dimension of the row space of a 3 x 5 matrix A is 2 d. What is the dimension of the solution space of the homogeneous system Ax = 0?

Dimension of the solution space of the homogeneous system = dimension of kernel=3.

Step-by-step explanation:

Given a matrix has 3 rows and 5 columns.

Dimension of Domain=Number of columns in the matrix=5d

Dimension of the row space = 2d

We know that dimension of row space = rank of matrix=2d

Rank-nullity theorem : Rank+nullity = dimension of domain=Number of columns in the matrix.

By using rank-nullity theorem

2+nullity=5

Nullity=5-2

Nullity=3

Dimension of kernel=3d

Dimension of kernel=Dimension of solution space

Dimension of solution space=3d

Hence, the dimension of solution space of the homogeneous system = 3d.