Use the information in the table to: (i) find the dimensions of the row space of

Question

Use the information in the table to: (i) find the dimensions of the row space of A, column space of A, null space of A, and null space of AT; (ii) determine whether or not the linear system Ax = b is consistent; (ili) find the number of parameters in the general solution of each system in (ii) that is consistent. Size of A Rank (A) Rank ([A |b) 5 5 x 5 The dimension of the row space of A is The dimension of the column space of A is The dimension of the null space of A is The dimension of the null space of A is

Explanation / Answer

Size of A = 5 x 5

Rank of A = 5

Rank of [ A I B ] = 5

(i)

Dimension of row space = no. of linearly independent row = 5 (As rank of 5x5 matrix is 5)

Dimension of column space = no. of linearly independent column = 5 (As rank of 5x5 matrix is 5)

Rank = dimension of Col (A) + dimension of null A

5 =5 + dimension of null A

Dimension of null A = 5-5 = 0

Rank = dimension of Row(A) + dimension of null AT

5 =5 + dimension of null AT

Dimension of null AT = 5-5 = 0

(ii)

Linear system is consistent as rank of A = rank of [A I b ] .

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