Suppose all the pivot variables come last instead of first. Describe all four blocks in

Which of these rules gives a correct definition of the rank of A? (a) The number of nonzero rows in R. (b) The number of columns minus the total number of rows. (c) The number of columns minus the number of free columns. (d) The number of l's in the matrix R.

Find the reduced row echelon forms R and the rank of these matrices: (a) The 3 by 4 matrix with all entries equal to 4. (b) The 3 by 4 matrix with aU = i + j - 1. (c) The 3 by 4 matrix with aU = (-I)j.

Find the reduced R for each of these (block) matrices: [ 0 0 0] A = 0 0 3 246 B=[A A] c

Suppose all the pivot variables come last instead of first. Describe all four blocks in the reduced echelon form (the block B should be r by r): R=[~ ~]. What is the nullspace matrix N containing the special solutions?

(Silly problem) Describe all 2 by 3 matrices A I and A2 , with row echelon forms RI and R2, such that RI + R2 is the row echelon form of Al + A2. Is is true that RI = Al and R2 = A2 in this case? Does RI - R2 equal rref(Al - A2)?

If A has r pivot columns, how do you know that AT has r pivot columns? Give a 3 by 3 example with different column numbers in pivcol for A and AT.

What are the special solutions to Rx = 0 and y T R = 0 for these R? R = [~ i ~~] R = [~ ~ ~]

Problems 8-11 are about matrices of rank r = 1.

Problems 8-11 are about matrices of rank r = 1.

Problems 8-11 are about matrices of rank r = 1.

Problems 8-11 are about matrices of rank r = 1.

Problems 12-14 are about r by r invertible matrices inside A.

Problems 12-14 are about r by r invertible matrices inside A.

Problems 12-14 are about r by r invertible matrices inside A.

Problems 15-20 show that rank(AB) is not greater than rank(A) or rank(B).

Problems 15-20 show that rank(AB) is not greater than rank(A) or rank(B).

Problems 15-20 show that rank(AB) is not greater than rank(A) or rank(B).

Problems 15-20 show that rank(AB) is not greater than rank(A) or rank(B).

Problems 15-20 show that rank(AB) is not greater than rank(A) or rank(B).

Problems 15-20 show that rank(AB) is not greater than rank(A) or rank(B).

Suppose A and B have the same reduced row echelon form R. (a) Show that A and B have the same nullspace and the same row space.

Express A and then B as the sum of two rank one matrices: rank = 2 A = [: 1 0] 1 4 1 8

Answer the same questions as in Worked Example 3.3 C for [ 1 1 2 2] A= 2 2 4 4 1 c 2 2 and

What is the nullspace matrix N (containing the special solutions) for A, B, C? A = [I I] and B = [~ ~ ] and C = [I I I].

Every m by n matrix o/rank r reduces to (m by r) times (r by n): Write the 3 by 4 matrix A in equation (1) at the start of this section as the product of the 3 by 2 matrix from the pivot columns and the 2 by 4 matrix from R.

Suppose A is an m by n matrix of rank r. Its reduced echelon form is R. Describe exactly the matrix Z (its shape and all its entries) that comes from transposing the reduced row echelon/arm 0/ R' (prime means transpose): R = rref(A) and Z = (rref(R'))'.

Suppose R is m by n of rank r, with pivot columns first: R = [~ ~]. (a) What are the'shapes of those four blocks? (b) Find a right-inverse B with RB = I if r = m. (c) Find a left-inverse C with CR = I if r = n. (d) What is the reduced row echelon form of RT (with shapes)? (e) What is the reduced row echelon form of RT R (with shapes)? Prove that RT R has the same nullspace as R. Later we show that AT A always has the same nullspace as A (a valuable fact)

Suppose you allow elementary column operations on A as well as elementary row operations (which get to R). What is the "row-and-column reduced form" for an m by n matrix of rank r?