A is an m Times n matrix Check the true statements below A I

A is an m Times n matrix. Check the true statements below: A. If the equation Ax = b is consistent, then Col A is R^m. B. The column space of A is the range of the mapping x rightarrow Ax. C. Col A is the set of all vectors that can be written as Ax for some x. D. The null space of A is the solution set of the equation Ax = 0. E. The kernel of a linear transformation is a vector space. F. The null space of an m Times n matrix is in R^m.

Solution

A.The statement is False. It will be True if the equation Ax = b is consistent

B. The statement is apparently True.

C. Ax equals a linear combination of columns of A using the x entries as weights. The statement is True.

D. By definition of Null(A), the statement is True.E.

E. The statement is True.The kernel of a linear transformation from a vector space V to a vector space W is a subspace of V.

F. The statement is False. The null space of a mxn matrix A is in Rⁿ and not R^m.