Linear algebra exam 3 terms

An m x n upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown in the matrix to the right. When is a square upper triangular matrix invertible? Justify your answer [3 4 7 4 0 1 4 6 0 0 2 8 0 0 0 1]

A square upper triangular matrix is invertible when all entries on its main diagonal are nonzero. If all of the entries on its main diagonal are nonzero, then the n × n matrix has n pivot positions

Mark each statement as true or false. Justify each answer. Here, A is an m x n matrix Each line in ℝ^n is a onedimensional subspace of ℝ^n

False, because any subspace of ℝ^n must contain the zerovector. Therefore, a line can only be a one-dimensional subspace of ℝ^n if it passes through the origin

For this exercise assume that the matrices are all n x n. Each part of this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Justify each answer. If A is an n × n matrix, then the equation Ax = b has at least one solution for each b in ℝ^n

False; by the Invertible Matrix Theorem Ax=b has at least one solution for each b in ℝ^n only if a matrix is invertible.

Mark each statement as true or false. Justify each answer. A subset H of ℝ^n is a subspace if the zero vector is in H

This statement is false. For each u and v in H and each scalar c, the sum u + v and the vector cu must also be in H.

Mark each statement as true or false. Justify each answer. The column space of a matrix A is the set of solutions of Ax = b

This statement is false. The column space of A is the set of all b for which Ax = b has a solution.

Mark each statement as true or false. Justify each answer. If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A

This statement is false. The columns of an echelon form of a matrix are often not in the column space of the original matrix.

Mark each statement as true or false. Justify each answer. The null space of an m × n matrix is a subspace of ℝ^n

This statement is true. For an m x n matrix A, the solutions of Ax= 0 are vectors in ℝ^n and satisfy the properties of a vector space.

Mark each statement as true or false. Justify each answer. Given vectors v1 , ...,vp in ℝ^n , the set of all linear combinations of these vectors is a subspace of ℝ^n

This statement is true. This set satisfies all properties of a subspace

Mark each statement as true or false. Justify each answer. Here, A is an m x n matrix If B={v1,...,vp} is a basis for a subspace H and if x=c1v1 + ... + cp vp, then c1,...,cp are the coordinates of x relative to the basis B.

True, because any coordinate in a subspace H, with basis B, can only be written in one way as a linear combination of basis vectors. The linear combination gives a unique coordinate vector [x]b that is composed of the coordinates of x relative to B.

Mark each statement as true or false. Justify each answer. Here, A is an m x n matrix If a set of p vectors spans a pdimensional subspace H of ℝ^n , then these vectors form a basis of H.

True, because if a set of p vectors spans a p-dimensional subspace H of ℝ^n , then these vectors must be linearly independent. Any linearly independent spanning set of p vectors forms a basis in p dimensions.

Mark each statement as true or false. Justify each answer. Here, A is an m x n matrix The dimensions of Col A and Nul A add up to the total number of columns in A

True, because the Rank Theorem states that if matrix A has n columns, then rank A + dim Nul A = n . Since rank A is the same as dim Col A, the dimensions of Col A and Nul A add up to the total number of columns in A

Mark each statement as true or false. Justify each answer. Here, A is an m x n matrix The dimension of Col A is the number of pivot columns in A.

True, because the pivot columns of A form a basis for Col A. Therefore, the number of pivot columns of A is the same as the dimension of Col A.

For this exercise assume that the matrices are all n x n. Each part of this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Justify each answer. If A^T is not invertible, then A is not invertible

True; by the Invertible Matrix Theorem if A^T is not invertible all statements in the theorem are false, including A is invertible. Therefore, A is not invertible.

For this exercise assume that the matrices are all n x n. Each part of this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Justify each answer. If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivot positions

True; by the Invertible Matrix Theorem if the equation Ax=0 has a nontrivial solution, then matrix A is not invertible. Therefore, A has fewer than n pivot positions.

For this exercise assume that the matrices are all n x n. Each part of this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Justify each answer. If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n × n identity matrix

True; by the Invertible Matrix Theorem if the equation Ax=0 has only the trivial solution, then the matrix is invertible. Thus, A must also be row equivalent to the n x n identity matrix

Suppose F is a 5 × 5 matrix whose column space is not equal to ℝ^5. What can you say about Nul F?

If Col F≠ℝ^5, then the columns of F do not span ℝ^5. Since F is square, the Invertible Matrix Theorem shows that F is not invertible and the equation Fx=0 has a nontrivial solution. Therefore, Nul F contains a nonzero vector

Suppose a 4 × 7 matrix A has four pivot columns. Is Col A = ℝ^4 ? Is Nul A = ℝ^3 ? Explain your answers. Is Nul A = ℝ^3?

No, because the null space of a 4 x 7 matrix is a subspace of ℝ^7 . Although dim Nul A = 3, it is not strictly equal to ℝ^3 because each vector in Nul A has seven components. Each vector in ℝ^3 has three components. Therefore, Nul A is isomorphic to ℝ^3 , but not equal

Suppose the columns of a matrix A = [a1, ..., ap] are linearly independent. Explain why {a1, ..., ap} is a basis for Col A

Since Col A is the set of all linear combinations of a1 , ..., ap , the set {a1, ...,ap} spans Col A. Because {a1, ..., ap} is also linearly independent, it is a basis for Col A.

Suppose the first two columns, b1 and b2, of B are equal. What can you say about the columns of AB (if AB is defined)? Why?

The first two columns of AB are Ab1 and Ab2 . They are equal since b1 and b2 are equal.

Mark each statement as true or false. Justify each answer. A subspace of ℝ^n is any set H such that (i) the zero vector is in H, (ii) u , v, and u + v are in H, and (iii) c is a scalar and cu is in H.

The statement is false. Conditions (ii) and (iii) must be satisfied for each u and v in H, which is not specified in the given statement.

Let A, B, and C be arbitrary matrices for which the indicated sums and products are defined. Mark each statement True or False. Justify each answer. (AB)C = (AC)B

The statement is false. The associative law of multiplication for matrices states that A(BC) = (AB)C.

Mark each statement as true or false. Justify each answer. The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of ℝ^m

The statement is false. The described set is the null space of an m × n matrix A. This set is a subspace of ℝ^n

Let A, B, and C be arbitrary matrices for which the indicated sums and products are defined. Mark each statement True or False. Justify each answer. If A and B are 3 × 3 matrices and B = [b1 b2 b3 , then AB = [Ab1 + Ab2 + Ab3]

The statement is false. The matrix [Ab1 + Ab2 + Ab3] is a 3x1 matrix, and AB must be a 3x3 matrix. The plus signs should be spaces between the 3 columns.

Let A, B, and C be arbitrary matrices for which the indicated sums and products are defined. Mark each statement True or False. Justify each answer. (AB)^T = A^T B^T

The statement is false. The transpose of the product of two matrices is the product of the transposes of the individual matrices in reverse order, or (AB)^T = B^T A^T

Mark each statement as true or false. Justify each answer. Row operations do not affect linear dependence relations among the columns of a matrix

The statement is true. If a series of row operations is performed on a matrix A to form B, then the equations Ax = 0 and Bx = 0 have the same set of solutions

Let A, B, and C be arbitrary matrices for which the indicated sums and products are defined. Mark each statement True or False. Justify each answer. The second row of AB is the second row of A multiplied on the right by B

The statement is true. Let row(i)(A) denote the ith row of matrix A. Then row(i)(AB) = row(i)(A)B. Letting i = 2 verifies this statement.

Mark each statement as true or false. Justify each answer. If v1 , ..., vp are in ℝ^n , then S= Span {v1 , ...,vp} is the same as the column space of the matrix A= [v1• • • vp]

The statement is true. The column space of A and S are both the set of all linear combinations of v1 , ..., vp

Mark each statement as true or false. Justify each answer. The columns of an invertible n × n matrix form a basis for ℝ^n

The statement is true. The columns of an invertible n x n matrix are linearly independent and span ℝ^n , so they form a basis for ℝ^n

Let A, B, and C be arbitrary matrices for which the indicated sums and products are defined. Mark each statement True or False. Justify each answer. The transpose of a sum of matrices equals the sum of their transposes

The statement is true. This is a generalized statement that follows from the theorem (A + B)^T = A^T + B^T

For this exercise assume that the matrices are all n x n. Each part of this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Justify each answer. If the columns of A span ℝ^n , then the columns are linearly independent

True; the Invertible Matrix Theorem states that if the columns of A span ℝ^n , then matrix A is invertible. Therefore, the columns are linearly independent.

Suppose a 4 × 7 matrix A has four pivot columns. Is Col A = ℝ^4 ? Is Nul A = ℝ^3 ? Explain your answers. Is Col A = ℝ^4?

Yes, because the column space of a 4 x 7 matrix is a subspace of ℝ^4. There is a pivot in each row, so the column space is 4-dimensional. Since any 4-dimensional subspace of ℝ^4 is ℝ^4 , Col A = ℝ^4