[M] Repeat Exercise 43 with the matrices A and B from

Problem 44E [M] Repeat Exercise 43 with the matrices A and B from Exercise 42. Then give an explanation for what you discover, assuming that B was constructed as specified. Exercise 43: [M] With A and B as in Exercise 41, select a column v of A that was not used in the construction of B and determine if v is in the set spanned by the columns of B. (Describe your calculations.) Exercise 41: [M] Use as many columns of A as possible to construct a matrix B with the property that the equation Bx = 0 has only the trivial solution. Solve Bx = 0 to verify your work. Exercise 42: [M] Use as many columns of A as possible to construct a matrix B with the property that the equation Bx = 0 has only the trivial solution. Solve Bx = 0 to verify your work.

Problem 1E In Exercises 1–4, determine if the vectors are linearly independent. Justify each answer.

Problem 2E In Exercises 1–4, determine if the vectors are linearly independent. Justify each answer.

Problem 4E In Exercises 1–4, determine if the vectors are linearly independent. Justify each answer.

Problem 5E In Exercises 5–8, determine if the columns of the matrix form a linearly independent set. Justify each answer.

Problem 3E In Exercises 1–4, determine if the vectors are linearly independent. Justify each answer.

Problem 6E In Exercises 5–8, determine if the columns of the matrix form a linearly independent set. Justify each answer.

Problem 7E In Exercises 5–8, determine if the columns of the matrix form a linearly independent set. Justify each answer.

Problem 9E In Exercises 9 and 10, (a) for what values of h is v3 in Span {v1, v2}, and (b) for what values of h is { v1, v2, v3} linearly dependent? Justify each answer.

Problem 10E In Exercises 9 and 10, (a) for what values of h is v3 in Span {v1, v2}, and (b) for what values of h is { v1, v2, v3} linearly dependent? Justify each answer.

Problem 8E In Exercises 5–8, determine if the columns of the matrix form a linearly independent set. Justify each answer.

Problem 11E In Exercises 11–14, find the value(s) of h for which the vectors are linearly dependent. Justify each answer.

Problem 12E In Exercises 11–14, find the value(s) of h for which the vectors are linearly dependent. Justify each answer.

Problem 13E In Exercises 11–14, find the value(s) of h for which the vectors are linearly dependent. Justify each answer.

Problem 14E In Exercises 11–14, find the value(s) of h for which the vectors are linearly dependent. Justify each answer.

Problem 15E Determine by inspection whether the vectors in Exercises 15–20 are linearly independent. Justify each answer.

Problem 16E Determine by inspection whether the vectors in Exercises 15–20 are linearly independent. Justify each answer.

Problem 17E Determine by inspection whether the vectors in Exercises 15–20 are linearly independent. Justify each answer.

Problem 19E Determine by inspection whether the vectors in Exercises 15–20 are linearly independent. Justify each answer.

Problem 20E Determine by inspection whether the vectors in Exercises 15–20 are linearly independent. Justify each answer.

Problem 18E Determine by inspection whether the vectors in Exercises 15–20 are linearly independent. Justify each answer.

Problem 21E In Exercises 21 and 22, mark each statement True or False. Justify each answer on the basis of a careful reading of the text. a. The columns of a matrix A are linearly independent if the equation Ax = 0 has the trivial solution. b. If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S. c. The columns of any 4 × 5 matrix are linearly dependent. d. If x and y are linearly independent, and if {x, y, z} is linearly dependent, then z is in Span {x, y}

Problem 25E In Exercises 23–26, describe the possible echelon forms of the matrix. Use the notation of Example 1 in Section 1.2. A is a 4 × 2 matrix, A = [a1 a2] and a2 is not a multiple of a1. Example 1: a. Determine if the set { v1, v2, v3,} is linearly independent. b. If possible, find a linear dependence relation among v1, v2 and v3,

Problem 24E In Exercises 23–26, describe the possible echelon forms of the matrix. Use the notation of Example 1 in Section 1.2. A is a 3 × 3 matrix with linearly independent columns. Example 1: a. Determine if the set { v1, v2, v3,} is linearly independent. b. If possible, find a linear dependence relation among v1, v2 and v3,

Problem 23E In Exercises 23–26, describe the possible echelon forms of the matrix. Use the notation of Example 1 in Section 1.2. A is a 2 × 2 matrix with linearly dependent columns. Example 1: a. Determine if the set { v1, v2, v3,} is linearly independent. b. If possible, find a linear dependence relation among v1, v2 and v3,

Problem 22E In Exercises 21 and 22, mark each statement True or False. Justify each answer on the basis of a careful reading of the text. a. If u and v are linearly independent, and if w is in Span {u, v} then {u, v, w} is linearly dependent. b. If three vectors in ?3 lie in the same plane in ?3, then they are linearly dependent. c. If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent. d. If a set in ?n is linearly dependent, then the set contains more than n vectors.

Problem 26E In Exercises 23–26, describe the possible echelon forms of the matrix. Use the notation of Example 1 in Section 1.2. A is a 4 × 3 matrix, A = [a1 a2 a3] such that {a1 a2} is linearly independent and a3 is not in Span {a1 a2}. Example 1: a. Determine if the set { v1, v2, v3,} is linearly independent. b. If possible, find a linear dependence relation among v1, v2 and v3,

Problem 27E How many pivot columns must a 6 × 4 matrix have if its columns are linearly independent? Why?

Problem 28E How many pivot columns must a 4 × 6 matrix have if its columns span ?4? Why?

Problem 30E a. Fill in the blank in the following statement: “If A is an m × n matrix, then the columns of A are linearly independent if and only if A has __________ pivot columns.” b. Explain why the statement in (a) is true.

Problem 29E Construct 3 × 2 matrices A and B such that Ax = 0 has a nontrivial solution, but Bx = 0 has only the trivial solution.

Problem 32E Exercises 31 and 32 should be solved without performing row operations. [Hint: Write Ax = 0 as a vector equation.] Given observe that the first column minus three times the second column equals the third column. Find a nontrivial solution of Ax = 0.

Problem 31E Exercises 31 and 32 should be solved without performing row operations. [Hint: Write Ax = 0 as a vector equation.] Given observe that the third column is the sum of the first two columns. Find a nontrivial solution of Ax = 0.

Problem 33E Each statement in Exercises 33–38 is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If a statement is true, give a justification. (One specific example cannot explain why a statement is always true. You will have to do more work here than in Exercises 21 and 22.) If v1,…, v2 are in ?4 and v3 = 2v1 + v2, then {v1 + v2 + v3 + v4} is linearly dependent.

Problem 34E Each statement in Exercises 33–38 is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If a statement is true, give a justification. (One specific example cannot explain why a statement is always true. You will have to do more work here than in Exercises 21 and 22.) If v1 and v2 are in ?4 and v2 is not a scalar multiple of v1,then {v1, v2} is linearly independent.

Problem 36E Each statement in Exercises 33–38 is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If a statement is true, give a justification. (One specific example cannot explain why a statement is always true. You will have to do more work here than in Exercises 21 and 22.) If v1, v2, v3 are in ?3 and v3 is not a linear combination of v1, v2, then {v1, v2, v3} is linearly independent

Problem 35E Each statement in Exercises 33–38 is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If a statement is true, give a justification. (One specific example cannot explain why a statement is always true. You will have to do more work here than in Exercises 21 and 22.) If v1,…, v5 are in ?5 and v3 = 0, then {v1 + v2 + v3 + v4 + v5} is linearly dependent.

Problem 40E Suppose an m × n matrix A has n pivot columns. Explain why for each b in ?m the equation Ax = b has at most one solution. [Hint: Explain why Ax = b cannot have infinitely many solutions.]

Problem 42E [M] In Exercises 41 and 42, use as many columns of A as possible to construct a matrix B with the property that the equation Bx = 0 has only the trivial solution. Solve Bx = 0 to verify your work.

Problem 43E [M] With A and B as in Exercise 41, select a column v of A that was not used in the construction of B and determine if v is in the set spanned by the columns of B. (Describe your calculations.) Exercise 41: [M] Use as many columns of A as possible to construct a matrix B with the property that the equation Bx = 0 has only the trivial solution. Solve Bx = 0 to verify your work.

Problem 41E [M] In Exercises 41 and 42, use as many columns of A as possible to construct a matrix B with the property that the equation Bx = 0 has only the trivial solution. Solve Bx = 0 to verify your work.

Problem 37E Each statement in Exercises 33–38 is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If a statement is true, give a justification. (One specific example cannot explain why a statement is always true. You will have to do more work here than in Exercises 21 and 22.) If v1,…, v4 are in ?4 and { v1, v2, v3} is linearly dependent, then { v1, v2, v3, v4} is also linearly dependent.

Problem 39E Suppose A is an m × n matrix with the property that for all b in ?m the equation Ax = b has at most one solution. Use the definition of linear independence to explain why the columns of A must be linearly independent.

Problem 38E Each statement in Exercises 33–38 is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If a statement is true, give a justification. (One specific example cannot explain why a statement is always true. You will have to do more work here than in Exercises 21 and 22.) If {v1,…, v4} is a linearly independent set of vectors in ?4, then { v1, v2, v3} is also linearly independent.