F.7 [ 25 points] Let A be an m × n matrix (m rows and n columns). Suppose you kn
ID: 3138766 • Letter: F
Question
F.7 [ 25 points] Let A be an m × n matrix (m rows and n columns). Suppose you know that the linear vector equation Ax 0 has k free variables. Please answer the following questions in terms of integers m, n, and k. (a) [5 points] What is the rank of the matrix A ? (b) [5 points ] What is the dimension of the row space ? (c) [5 points] What is the dimension of the column space ? (d) [ 5 points] What is the dimension of the kernel (= null space) of A ? (e) [5 points] W hat is the dimension of the cokernel(null space of transpose matrix AT) of A ? F.8 [10 points] For the matrix A 2-5 3,find cofactor of the element in the secontd -1 -31 row and third column.Explanation / Answer
7. (a). If the linear matrix equation AX = 0 has k free variables, then null(A), which is the set of solutions to the equation AX = 0, has dimension k. Then, as per the rank-nullity theorem, rank(A) = no. of columns of A–nullity of A = n-k.
(b). dim(row(A)) = rank(A) = n-k.
(c ). dim(col(A)) = dim(row(A)) = n-k.
(d). dim(ker(A)) = dim(null(A)) = k.
(e). null (AT) has dimension m- rank(A) = m-k.
Please post the other question again separately.
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