Question

Verify the rank theorem: 1. dim(ColA1)=?dim( Row A1)= 2. rank(A1)+dim(NulA1)= ? ans =3 Verify the rank theorem: 1. dim(ColA2)=?dim(RowA2)= 2. rank(A2)+dim(NulA2)= ? ans =5 Verify the rank theorem: 1. dim(ColA3)= ? dim( Row A3)= 2. rank(A3)+dim( Nul A3)= ?

Answer 1

The rank theorem states that the dimension of the column space is equal to the dimension of the row space, and the rank of a matrix is equal to the dimension of its column space plus the dimension of its null space. These principles can be verified using the given equations.

Step-by-step explanation:

The rank theorem, also known as the fundamental theorem of linear algebra, states the following:

1. The dimension of the column space (ColA) of a matrix A is equal to the dimension of the row space (RowA). In this case, we have dim(ColA1) = dim(RowA1).
2. The rank of a matrix A (rank(A)) is equal to the dimension of its column space plus the dimension of its null space (NulA). Here, we have rank(A1) + dim(NulA1) = 3.

We can apply the same reasoning to the second and third statements. Based on the given equation, we have dim(ColA2) = dim(RowA2), and rank(A2) + dim(NulA2) = 5. Similarly, we have dim(ColA3) = dim(RowA3), and rank(A3) + dim(NulA3) = ? (unspecified).