Linear Algebra show all work If a 3 x 6 matrix A has rank 2

(Linear Algebra, show all work) If a 3 x 6 matrix A has rank 2, find dim Null A, dim Row A, and rank A^T.

Solution

Given Matrix A has m rows of 3, n colums of 6 and Rank (r) of 2.

so vbelow are answers of questions-

1.The Fundamental Theorem finds the dimensions of the four subspaces. One fact stands out: The row space and column space have the same dimension r (the rank of the matrix).

A has Rank (r) =2

Nullspace N(A) = n-r = 6-2 =4 i.e. Null(A) = 4

2. The first two rows are a basis. The row space contains combinations of all three rows, but the third row (the zero row) adds nothing new. So rows 1 and 2 span the row space of matrix A.

The dimension of the row space is the rank r. The nonzero rows of A form a basis.

So Dimention of Row A = 2

3. Basic Algebra therom :- If A is m by n of rank r, its left nullspace has dimension m-r.

So The left nullspace of A (the nullspace of AT) has dimension m - r i.e. 3-2 = 1