Introduction to Linear Algebra 4th Edition solutions

Introduction to Linear Algebra 4
(a) If a 7 by 9 matrix has rank 5, what are the dimensions of the four subspaces? What is the sum of all four dimensions?

Introduction to Linear Algebra 4
Find bases and dimensions for the four subspaces associated with A and B: A = [1 2 4] 248 [ 1 2 4] and B = 2 5 8 .

Introduction to Linear Algebra 4
Find a basis for each of the four subspaces associated with A:

Introduction to Linear Algebra 4
Construct a matrix with the required property or explain why this is impossible: (a) Column space contains U J.[i J. row space contains U], U]' (b) Column space has basis [i J, nullspace has basis [} J. (c) Dimension of nullspace = 1 + dimension of left nUllspace. (d) Left nullspace contains [~], row

Introduction to Linear Algebra 4
If V is the subspace spanned by (1,1,1) and (2,1,0), find a matrix A that has V as its row space. Find a matrix B that has V as its nullspace.

Introduction to Linear Algebra 4
Without elimination, find dimensions and bases for the four subspaces for [ 0 3 3 3] [1] A = 0 0 0 0 and B = 4 . o 1 0 1 5

Introduction to Linear Algebra 4
Suppose the 3 by 3 matrix A is invertible. Write down bases for the four subspaces for A, and also for the 3 by 6 matrix B = [A A]

Introduction to Linear Algebra 4
What are the dimensions of the four subspaces for A, B, and C, if I is the 3 by 3 identity matrix and 0 is the 3 by

Introduction to Linear Algebra 4
Which subspaces are the same for these matrices of different sizes? (a) [A] and [~] (b) [ ~] and [~ ~ ] . Prove that all three of those matrices have the same rank r.

Introduction to Linear Algebra 4
If the entries of a 3 by 3 matrix are chosen randomly between 0 and 1, what are the most likely dimensions of the four subspaces? What if the matrix is 3 by 5?

Introduction to Linear Algebra 4
(Important) A is an m by n matrix of rank r. Suppose there are right sides b for which Ax = b has no solution. (a) What are all inequalities or <) that must be true between m, n, and r? (b) How do you know that AT y = 0 has solutions other than y = O?

Introduction to Linear Algebra 4
Construct a matrix with (1,0,1) and (1,2,0) as a basis for its row space and its column space. Why can't this be a basis for the row space and nullspace?

Introduction to Linear Algebra 4
True or false (with a reason or a counterexample): (a) If m = n then the row space of A equals the column space. (b) The matrices A and -A share the same four subspaces. (c) If A and B share the same four subspaces then A is a multiple of B

Introduction to Linear Algebra 4
Without computing A, find bases for its four fundamental subspaces: [ L 0 0] [1 2 3 4] A= 6 1 0 0 1 2 3 . 9810012

Introduction to Linear Algebra 4
If you exchange the first two rows of A, which of the four subspaces stay the same? If v = (1,2,3,4) is in the left nullspace of A, write down a vector in the left nullspace of the new matrix.

Introduction to Linear Algebra 4
Explain why v = (1,0, -1) cannot be a row of A and also in the nUlispace.

Introduction to Linear Algebra 4
Describe the four subspaces of R3 associated with '" [0 1 0] A = 0 0 1 000 and I + A = [~ i !l

Introduction to Linear Algebra 4
(Left nullspace) Add the extra column b and reduce A to echelon form: [ 1 2 3 bI ] [A b] = 4 5 6 b2 7 8 9 b3 [ 1 2 3 -3 -6 000 A combination of the rows of A has produced the zero row. What combination is it? (Look at b3 - 2b2 + bi on the right side.) Which vectors are in the nullspace of AT and whi

Introduction to Linear Algebra 4
Following the method of Problem 18, reduce A to echelon form and look at zero rows. The b column tells which combinations you have taken of the rows: [ 1 2 (a) 3 4 4 6 (b) 1 2 2 3 2 4 2 5 From the b column after elimination, read off m - r basis vectors in the left nUllspace. Those y's are combinatio

Introduction to Linear Algebra 4
(a) Check that the solutions to Ax = 0 are perpendicular to the rows: [ 1 0 0] [4 2 0 1] A = 2 1 0 0 0 1 3 = ER. 3410000 (b) How many independent solutions to AT y = O? Why is y T the last row of E- 1?

Introduction to Linear Algebra 4
Suppose A is the sum of two matrices of rank one: A = uv T + w z T. (a) Which vectors span the column space of A? (b) Which vectors span the row space of A? (c) The rank is less than 2 if or if __ (d) Compute A and its rank if u = z = (1,0,0) and v = w = (0,0,1).

Introduction to Linear Algebra 4
Construct A = uvT + wzT whose column space has basis (1,2,4), (2,2,1) and whose row space has basis (1,0), (1,1). Write A as (3 by 2) times (2 by 2).

Introduction to Linear Algebra 4
Without mUltiplying matrices, find bases for the row and column spaces of A: [ 1 2] [3 0 3] A= ~ ~ 1 1 2 . How do you know from these shapes that A cannot be invertible?

Introduction to Linear Algebra 4
Important) AT y = d is solvable when d is in which of the four subspaces? The solution y is unique when the contains only the zero vector.

Introduction to Linear Algebra 4
True or false (with a reason or a counterexample): (a) A and AT have the same number of pivots. (b) A and AT have the same left nullspace. (c) If the row space equals the column space then AT = A. (d) If AT = -A then the row space of A equals the column space.

Introduction to Linear Algebra 4
(Rank of A B) If A B = C, the rows of C are combinations of the rows of __ So the rank of C is not greater than the rank of . Since BT AT = CT, the rank of C is also not greater than the rank of __

Introduction to Linear Algebra 4
If a, b, c are given with a i= 0, how would you choose d so that [~ ~ ] has rank I? Find a basis for the row space and nUllspace. Show they are perpendicular!

Introduction to Linear Algebra 4
Find the ranks of the 8 by 8 checkerboard matrix B and the chess matrix C: 1 0 1 0 1 0 1 0 r n b q k b n r 0 1 0 1 0 1 0 1 P P P P P P P P B= 1 0 1 0 1 0 1 0 and C= four zero rows p p p p p p p p 0 1 0 1 0 1 0 1 r n b q k b n r The numbers r, n, b, q, k, p are all different. Find bases for the row sp

Introduction to Linear Algebra 4
Can tic-tac-toe be completed (5 ones and 4 zeros in A) so that rank (A) = 2 but neither side passed up a winning move?

Introduction to Linear Algebra 4
Can tic-tac-toe be completed (5 ones and 4 zeros in A) so that rank (A) = 2 but neither side passed up a winning move?

Introduction to Linear Algebra 4
M is the space of 3 by 3 matrices. Multiply every matrix X in M by (a) Which matrices X lead to AX = zero matrix? (b) Which matrices have the form AX for some matrix X? (a) finds the "nullspace" of that operation AX and (b) finds the "column space". What are the dimensions of those two subspaces of M

Introduction to Linear Algebra 4
Suppose the m by n matrices A and B have the same four subs paces. If they are both in row reduced echelon form, prove that F must equal G: A=[~ ~] B=[~ ~]