Problem 13. (1 point) Problem 19. (1 paint) Are the following statements true or false? Let A = 4 5 and w = 7 1. If A is a 7 x 5 matrix, and I(x) - Ar is a linear transfor- Is win Cal(A)? Type "yes" or "no". mation, then I can be one-to one. 72. If mef(A ) = ref( B), then row(A) = now( #). Is win Nul(A)? Type "yes" or "no"_ 7 3. If A is a matrix, then the dimension of the row space of A is equal to the dimension of the column space of A. 7 4. If ref(A) = ref( B), then col(A) = col( B). Problem 20. (1 point) Problem 14. (1 point) A is an w X n matrix. Find the mull space for A= Check the true statements below: What is mull(A)? . A. The null space of an m x a matrix is in R.". .B. Col(A) is the set of all vectors that can be written as Ax for some I. . A. {} . C. The null space of A is the solution set of the equation Ar =0. . B. span 2 D. The column space of A is the range of the mapping J-+Ar . C. span . E. If the equation Ar = b is consistent, then Col(A) is R". . D. R. . F. The kernel of a linear transformation is a vector space. . E. spam Problem 15. (1 point) . F. span Suppose A is a 13 x 4 matrix and that T(x) = Ar. IF Y is one-to- one, find the dimension of the null space of A. . G. spams 2']} The nullity(A) = . H. none of the above Problem 16. (1 point) Suppose that A is an 11 x 5 matrix, and that # is an equivalent matrix in echelon form. If the number of pivot columns of & is 2, find the nullity of A. Problem 21. (1 paint) Let W be the set of all vectors of the form The nullity of A is _ 4r-45 -2 Problem 17. (1 point) Ar - (5r + 5) Sr + 3r Suppose that A is a 12 x 9 matrix. If the dimension of the column space of A is 4, then the dimension of the row space of A is _ 41 - (2r + 2x) with r, s and / real. Find a matrix A such that W = Col(A). Problem 18. (1 point) Suppose that A is a 5 x7 matrix that has an echelon form with one zero row. Find the dimension of the row space of A. the dimension of the column space of A, and the dimension of the null space of A. 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 _