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17 March, 08:01

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?

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  1. 17 March, 11:19
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    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.
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