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mathematics
linear algebra and its applications

Questions and Answers of Linear Algebra And Its Applications

Construct a nonzero 3 x 3 matrix A and a nonzero vector b such that b is in Nul A.
What can you say about the shape of an m x n matrix A when the columns of A form a basis for Rm?
Determine the values of the parameter s for which the system has a unique solution, and describe the solution. 38x1 + 5x2 = 3 12x₁ + 5sx₂ = 2
Find the determinants where, a d g b e h C f = 7. i
Letand B = {b1,b2}. Use the figure to estimate [w]B and [x]B. Confirm your estimate of [x] by using it and {b1,b2} to compute x. b₁ - [8]·₂-[-] = [-2] - [1 W X = 2 W
Letand B = {b1, b2}.Use the figure to estimate [X]B, [y]B, and [z]B. Confirm your estimates of [y]B and [Z]B by using them and {b1,b2} to compute y and z. b₁ b = [9] - [3] x [²³] x
The vector x is in a subspace H with a basis B = {b1,b2}. Find the B-coordinate vector of x. b₁ 1,b₂ = 2 -6 7 8 4 0 --[ X =
LetandDetermine if u is in the subspace of R4 generated by {v1, v2, v3}. VI V₁ = -2 4 3 V2 = 4 -7 9 7 درا = 5 -8 6 5
With A and p as in Exercise 7, determine if p is in Nul A.Data from in Exercise 7Letand A = [v1 v2 v3]. VI = -3 HCHD V2 = 8 V3 = -7 2 -8 6 -4 6 -7
LetandDetermine if p is in Col A, where A = [v1 v2 v3]. V₁ = 0 6 V2 = -2 2 3 V3 = 0 -6 3
Letand A = [v1 v2 v3].a. How many vectors are in {V1, V2, V3}? b. How many vectors are in Col A? c. Is p in Col A? Why or why not? VI = -3 HCHD V2 = 8 V3 = -7 2 -8 6 -4 6 -7
With u = (-2, 3, 1) and A as in Exercise 8, determine if u is in Nul A.Data from in Exercise 8LetandDetermine if p is in Col A, where A = [v1 v2 v3]. V₁ = 0 6 V2 = -2 2 3 V3 = 0 -6 3
Display a matrix A and an echelon form of A. Find bases for Col A and Nul A, and then state the dimensions of these subspaces. A = 1 -3 2 -4 -3 9 -6 2-4 -1 5 4-3 7 12 2 2 1 -3 2-4 0 0 5 -7 0 0 0
Display a matrix A and an echelon form of A. Find bases for Col A and Nul A, and then state the dimensions of these subspaces. A = 1 2 -3 3 1 0 0 0 2 1 0 0 2 5 -5 -8 -9 9 10 -7 -5
Justify each answer.(T/F) An elementary n x n matrix has either n or n + 1 nonzero entries.
Display a matrix A and an echelon form of A. Find bases for Col A and Nul A, and then state the dimensions of these subspaces. A
Justify each answer.(T/F) The transpose of an elementary matrix is an elementary matrix.
Give integers p and q such that Nul A is a subspace of RP and Col A is a subspace of Rq. A = 1 4 -5 -1 2 7 2 25 5 37 3 0 11
Give integers p and q such that Nul A is a subspace of RP and Col A is a subspace of Rq. A = 3 -9 9 1-5 1 7 2-5 1 2 -4
Find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace? -3 2 -4 9 -6 12 2 4 2 -4 5 -3
Justify each answer.(T/F) An elementary matrix must be square.
Find the matrix C whose inverse is C-1 4 6 7
For A as in Exercise 11, find a nonzero vector in Nul A and a nonzero vector in Col A.Data from in Exercise 11 A = 3 -9 9 1-5 1 7 2-5 1 2 -4
Give the 4 x 4 matrix that rotates points in R³ about the x-axis through an angle of 60°. (See the figure.) N. €3 e2
Assume that the matrices mentioned below have appropriate sizes. Justify each answer.(T/F) If A is a 3 x 3 matrix with three pivot positions, there exist elementary matrices E1,...,Ep such that
Assume that the matrices mentioned below have appropriate sizes. Justify each answer.(T/F) If AB = I, then A is invertible.
Assume that the matrices mentioned below have appropriate sizes. Justify each answer.(T/F) If A and B are square and invertible, then AB is invertible, and (AB)-1 = A-1B-1.
Assume that the matrices mentioned below have appropriate sizes. Mark each statement True or False (T/F). Justify each answer.(T/F) If AB = BA and if A is invertible, then A-1B = BA-1.
Suppose a 5 x 8 matrix A has two pivot columns. Is Col A = R2? What is the dimension of Nul A? Explain your answers.
Suppose A is invertible. Explain why ATA is also invertible. Then show that A-1 = (ATA)-1 AT.
Justify each answer. Here A is an m x n matrix.(T/F) If B = {v1,..., Vp) is a basis for a subspace H and if x = C1V1 + ... + Cpvp, then C1,..., Cp are the coordinates of x relative to the basis B.
LetDetermine P and Q as in Exercise 27, and compute Px and Qx. The figure shows that Qx is the reflection of x through the x1x2-plane.A Householder reflection through the plane x3 = 0.Data from in
Justify each answer.(T/F) A subspace of Rn is any set H such that (i) the zero vector is in H, (ii) u, v, and u + v are in H, (iii) c is a scalar and cu is in H.
Display a matrix A and an echelon form of A. Find a basis for Col A and a basis for Nul A. A = -3 9-2-7 2-6 4 8 ~ 3-9-2 2 1 0 0 -3 6 0 4 000 9 5
Display a matrix A and an echelon form of A. Find a basis for Col A and a basis for Nul A. 4 A = 6 554 5 3 4 9 -2 12 1 18 8 -3 - [ 0 0 2 1 0 6-5 5-6 0 0
Let A = LU, where L is an invertible lower triangular matrix and U is upper triangular. Explain why the first column of A is a multiple of the first column of L. How is the second column of A related
Justify each answer. Here A is an m x n matrix.(T/F) The dimension of the column space of A is rank A.
Certain dynamical systems can be studied by examining powers of a matrix, such as those below. Determine what happens to Ak and Bk as k increases (for example, try k = 2,...,16). Try to identify
Display a matrix A and an echelon form of A. Find a basis for Col A and a basis for Nul A. A = 2 3 نا -2 -5 -2 3 −1 0 2 0 0 0 0 -1 2 I 7 -2 3 9 6 6 7 0 4 0 0 3 9 7 5 3 4 3 7 6 0 3 1 1 0 0
Justify each answer. Here A is an m x n matrix.(T/F) If H is a p-dimensional subspace of Rn, then a linearly independent set of p vectors in H is a basis for H.
Justify each answer.(T/F) The columns of an invertible n x n matrix form a basis for Rn.
Display a matrix A and an echelon form of A. Find a basis for Col A and a basis for Nul A. A = 2 1 -1 -2 3 1 0 2 0 0 0 0 42 4 4226 8 550 5 0 0 8699 8 7 دی دی ان ای -7 4 5 -2 -5 0 5 0 -1 1
Given u in Rn with uTu = 1, let P = uuT (an outer product) and Q = I - 2P. Justify statements, (a) P2 = P(b) PT = P(c) Q2 = 1The transformation x ↦ Px is called a projection, and x
Justify answer or construction. If the subspace of all solutions of Ax = 0 has a basis consisting of three vectors and if A is a 5 x 7 matrix, what is the rank of A?
Justify each answer.(T/F) The column space of a matrix A is the set of solutions of Ax = b.
Justify answer or construction.If the rank of a 7 x 6 matrix A is 4, what is the dimension of the solution space of Ax = 0?
Justify each answer.(T/F) If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A.
If possible, construct a 3 x 4 matrix A such that dim Nul A = 2 and dim Col A = 2.
Let A be an n x n matrix such that the sum of the entries of each row equals zero. Explain why we can conclude that A is singular.
Show that a set of vectors {V1, V2,...,V5} in Rn is linearly dependent when dim Span {V1, V2,...,V5} = 4.
Let A be a 6 x 4 matrix and B a 4 x 6 matrix. Show that the 6 x 6 matrix AB cannot be invertible.
Suppose A is a 5 × 3 matrix and there exists a 3 × 5 matrix C such that CA = I3. Suppose further that for some given b in R5, the equation Ax = b has at least one solution. Show that this solution
Construct a 4 x 3 matrix with rank 1.
Construct bases for the column space and the null space of the given matrix A. Justify your work. A: = 3 -5 -7 -5 3 ܝ ܩ ܢ ܢܐ 7 -7 0 -4 -2 -3 -1 9 5 4 3 -11 -7 0
Construct bases for the column space and the null space of the given matrix A. Justify your work. A = 5 4 5 -8 2 1 1 -5 0 -8 2 -8 3 6 ∞ in ∞ 5 8 -8 -9 19 5
Let A be an n x p matrix whose column space is p-dimensional. Explain why the columns of A must be linearly independent.
Show how to use the condition number of a matrix A to estimate the accuracy of a computed solution of Ax = b. If the entries of A and b are accurate to about r significant digits and if the condition
Let An be the n x n matrix with 0's on the main diagonal and 1's elsewhere. Compute An-1 for n = 4, 5, and 6, and make a conjecture about the general form of An-1 for larger values of n.
Suppose columns 1, 3, 5, and 6 of a matrix A are linearly independent (but are not necessarily pivot columns) and the rank of A is 4. Explain why the four columns mentioned must be a basis for the
Construct a nonzero 3 x 3 matrix A and a nonzero vector b such that b is in Col A, but b is not the same as any one of the columns of A.
Suppose vectors b1,...,bp span a subspace W, and let {a1,...,aq} be any set in W containing more than p vectors. Fill in the details of the following argument to show that {a,...,aq} must be linearly
Compute the determinants using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column. 0 5 2 4 -3 4 1 0 1
Suppose the columns of a matrix A = [a1 · · · ap] are linearly independent. Explain why {a1,..., ap} is a basis for Col A.
Use Cramer’s rule to compute the solutions of the systems. 6x₁ + x₂ = 3 5x1 + 2x2 = 4
Use Cramer’s rule to compute the solutions of the systems. 3x1 - 2x2 = 3 -4x1 + 6x₂ = -5 ||
If Q is a 4 x 4 matrix and Col Q = R4, what can you say about solutions of equations of the form Qx = b for b in R4?
If P is a 5 x 5 matrix and Nul P is the zero subspace, what can you say about solutions of equations of the form Px = b for b in R5?
Illustrates a property of determinants. State the property. 3-6 9 3 1 5 3 -5 = 33 1 ل لیا 1-2 3 3 5 -5 3 3
Illustrates a property of determinants. State the property. 1 0 2 2 2 3-4 7 4 = 1 0 0 2 2 3-4 3 0
If R is a 6 x 6 matrix and Nul R is not the zero subspace, what can you say about Col R?
What can you say about Nul B when B is a 5 x 4 matrix with linearly independent columns?
Justify each answer. Assume that all matrices here are square.(T/F) If A is a 2 x 2 matrix with a zero determinant, then one column of A is a multiple of the other.
Justify each answer. Assume that all matrices here are square.(T/F) If two rows of a 3 x 3 matrix A are the same, then det A = 0.
Compute the determinants using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column. ล U 1 -2 -
Use Cramer’s rule to compute the solutions of the systems. -5x1 + 2x2 = 3xi - X2 = 9 -4
Justify each answer. Assume that all matrices here are square.(T/F) If A is a 3 x 3 matrix, then det 5 A = 5 det A.
Illustrates a property of determinants. State the property. 1 3-4 2 0 -3 3 -5 2 1 3-4 0-6 5 3-5 2
Compute the determinants using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column. | 1 3 2 12 1 4 4 1 2
Compute the determinants using a cofactor expansion across the first row. 4 1 7 5-8 0 3 926
Use Cramer’s rule to compute the solutions of the systems. x1 X1 + 3x2 + x3 = 8 -X1 + 3x1 + X2 |||| 2х3 = 4 = 4
Justify each answer. Assume that all matrices here are square.(T/F) If A and B are n x n matrices, with det A = 2 and det B = 3, then det(A + B) = 5.
Use Cramer’s rule to compute the solutions of the systems. x₁ + x₂ -5x1 || 2 0 +4x3 = X2 X3 = -1
Find the determinants by row reduction to echelon form. 15-4 -1 -4 -2-8 57
Find the determinants by row reduction to echelon form. 3-6 3-5 3-4 6 9 8
Determine the values of the parameter s for which the system has a unique solution, and describe the solution. 2sx1 + 5x2 = 8 6x1 +38x2 = 4
Find the determinants by row reduction to echelon form. 1 3 دیا Գ ԱՄՈՒԼՈ 1 -1 8722 0 1 -3
Compute the determinants using a cofactor expansion across the first row. 6-3 0 3 5 -5 258 -7
Compute the determinants using a cofactor expansion across the first row. 96 9 3 23
Compute the determinants using a cofactor expansion across the first row. 1 0 3-2 443 4 2 3 5
Find the determinants by row reduction to echelon form. 13 0 -2 -6 1 0 -1 0 1 -2 -1 2 3 2 5 5-6 2-4 -2 -3 10 -3 9
Find the determinants by row reduction to echelon form. 1 2 -3 -3 015 -4 -9 2 5 4 6 7 -14 0 7
Justify each answer. Assume that all matrices here are square.(T/F) If A is n x n and det A = 2, then det A3 = 6.
Determine the values of the parameter s for which the system has a unique solution, and describe the solution. SX₁ + 25x₂ = −1 3x1 + 6sx2 4
Find the determinants by row reduction to echelon form. 1 -1 0 1 0 3 -3 -1 -3 5 5 -2 0 0 4 3 3
Justify each answer. Assume that all matrices here are square.(T/F) If B is produced by interchanging two rows of A, then det B = det A.
Compute the determinants by cofactor expansions. At each step, choose a row or column that involves the least amount of computation. 7 -0% 6 0 7 4 8 0 9 0 4635
Determine the values of the parameter s for which the system has a unique solution, and describe the solution. SX₁ - 2x₂ = 1 4sx₁ + 4sx2 = 2
Justify each answer. Assume that all matrices here are square.(T/F) If B is produced by multiplying row 3 of A by 5, then det B = 5 det A.
Justify each answer. Assume that all matrices here are square.(T/F) If B is formed by adding to one row of A a linear of the other rows, then det B = det A. combination
Compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.Data from in Theorem 8 11-2 1 -1 -1 0 3 3

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