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mathematics
linear algebra
Linear Algebra 1st Edition Jim Hefferon - Solutions
There has been much interest in whether industries in the United States are moving from the Northeast and North Central regions to the South and West, motivated by the warmer climate, by lower wages, and by less unionization. Here is the transition matrix for large firms in Electric and Electronic
Here is a model of some kinds of learning The learner starts in an undecided state sU. Eventually the learner has to decide to do either response A (that is, end in state sA) or response B (ending in sB). However, the learner doesn't jump right from undecided to sure that A is the correct thing to
A certain town is in a certain country (this is a hypothetical problem). Each year ten percent of the town dwellers move to other parts of the country. Each year one percent of the people from elsewhere move to the town. Assume that there are two states sT , living in town, and sC, living
For the World Series application, use a computer to generate the seven vectors for p = 0.55 and p = 0:6. (a) What is the chance of the National League team winning it all, even though they have only a probability of 0.45 or 0.40 of winning any one game? (b) Graph the probability p against the
We define a transition matrix to have each entry nonnegative and each column sum to 1. (a) Check that the three transition matrices shown in this Topic meet these two conditions. Must any transition matrix do so? (b) Observe that if is a transition matrix from Show that a power of a transition
These questions refer to the coin-flipping game. (a) Check the computations in the table at the end of the first paragraph. (b) Consider the second row of the vector table. Note that this row has alternating 0's. Must p1(j) be 0 when j is odd? Prove that it must be, or produce a counterexample. (c)
Decide if each of these is an orthonormal matrix.(a)(b) (c)
Write down the formula for each of these distance-preserving maps.(b) The map that reflects about the line y = 2x (c) The map that reflects about y = -2x and translates over 1 and up 1
(a) The proof that a map that is distance-preserving and sends the zero vector to itself incidentally shows that such a map is one-to-one and onto (the point in the domain determined by d0, d1, and d2 corresponds to the point in the codomain determined by those three). Therefore any
(a) Verify that the properties described in the second paragraph of this Topic as invariant under distance-preserving maps are indeed so. (b) Give two more properties that are of interest in Euclidean geometry from your experience in studying that subject that are also invariant under
Find the line of best fit for the men's 1500 meter run. How does the slope compare with that for the men's mile? (The distances are close; a mile is about 1609 meters.)
Find the line of best fit for the records for women's mile.
Do the lines of best fit for the men's and women's miles cross?
In a highway restaurant a trucker told me that his boss often sends him by a roundabout route, using more gas but paying lower bridge tolls. He said that New York State calibrates the toll for each bridge across the Hudson, playing off the extra gas to get there from New York City against a lower
When the space shuttle Challenger exploded in 1986, one of the criticisms made of NASA's decision to launch was in the way they did the analysis of number of O-ring failures versus temperature (O-ring failure caused the explosion). Four O-ring failures would be fatal. NASA had data from 24 previous
This table lists the average distance from the sun to each of the first seven planets, using Earth's average as a unit.(a) Plot the number of the planet (Mercury is 1, etc.) versus the distance. It does not look like a line, and so finding the line of best fit is not fruitful. (b) It does, however
Let h: R2 ! R2 be the transformation that rotates vectors clockwise by π/4 radians. (a) Find the matrix H representing h with respect to the standard bases. Use Gauss's Method to reduce H to the identity. (b) Translate the row reduction to a matrix equation TjTj-1 . . . T1H = I (the prior item
Show that the equation of a line in R2 thru (x1, y1) and (x2, y2) is given by this determinant.
Many people know this mnemonic for the determinant of a 3 × 3 matrix: first repeat the first two columns and then sum the products on the forward diagonals and subtract the products on the backward diagonals. That is, first writeand then calculate this. h1,1h2,2h3,3 + h1,2h2,3h3,1 +
The cross product of the vectorsis the vector computed as this determinant. The first row's entries are vectors, the vectors from the standard basis for R3. Show that the cross product of two vectors is perpendicular to each vector.
Prove that each statement holds for 2 × 2 matrices. (a) The determinant of a product is the product of the determinants det(ST) = det(S) ∙ det(T). (b) If T is invertible then the determinant of the inverse is the inverse of the determinant det(T-1) = (det(T))-1. Matrices T and T′ are similar
Prove that the area of this region in the planeis equal to the value of this determinant. Compare with this.
(a) Find the 1 × 1, 2 × 2, and 3 × 3 matrices with i, j entry given by (-1)i+j. (b) Find the determinant of the square matrix with i, j entry (-1)i+j.
Is the determinant function linear-is det(x ∙ T + y ∙ S) = x ∙ det(T) + y ∙ det(S)?
Show that determinant functions are not linear by giving a case where |A+B| ≠ |A| + |B|.
Which real numbers θ makesingular? Explain geometrically.
If a third order determinant has elements 1, 2, . . . , 9, what is the maximum value it may have?
Refer to the definition of elementary matrices in the Mechanics of Matrix Multiplication subsection. (a) What is the determinant of each kind of elementary matrix? (b) Prove that if E is any elementary matrix then |ES| = |E||S| for any appropriately sized S. (c) (This question doesn't involve
Prove that the determinant of a product is the product of the determinants |TS| = |T| |S| in this way. Fix the n × n matrix S and consider the function d: Mn×n → R given by T → |TS|/|S|.(a) Check that d satisfies condition (1) in the definition of a determinant function.(b) Check condition
A submatrix of a given matrix A is one that we get by deleting some of the rows and columns of A. Thus, the first matrix here is a submatrix of the second.Prove that for any square matrix, the rank of the matrix is r if and only if r is the largest integer such that there is an r × r
Prove that a matrix with rational entries has a rational determinant.
Find the element of likeness in(a) Simplifying a fraction,(b) Powdering the nose,(c) Building new steps on the church,(d) Keeping emeritus professors on campus,(e) Putting B, C, D in the determinant
Use Gauss's Method to find each determinant.
Use Gauss's Method to find each.
For which values of k does this system have a unique solution? x + z - w = 2 y - 2z = 3 x + kz = 4 z - w = 2
Express each of these in terms of |H|.(a)(b)(c)
(a) Find the 1 × 1, 2 × 2, and 3 × 3 matrices with i, j entry given by i + j. (b) Find the determinant of the square matrix with i, j entry i + j.
Prove that the determinant of any triangular matrix, upper or lower, is the product down its diagonal.
Compute these both with Gauss's Method and the permutation expansion formula.This summarizes our notation for the 2- and 3- permutations.
What is the smallest number of zeros, and the placement of those zeros, needed to ensure that a 4 × 4 matrix has a determinant of zero?
We can divide a matrix into blocks, as here,which shows four blocks, the square 2×2 and 1×1 ones in the upper left and lower right, and the zero blocks in the upper right and lower left. Show that if a matrix is such that we can partition it as where J and K are square,
Prove that for any n × n matrix T there are at most n distinct reals r such that the matrix T - rI has determinant zero.
The nine positive digits can be arranged into 3 × 3 arrays in 9! ways. Find the sum of the determinants of these arrays.
Use the permutation expansion formula to derive the formula for 3 × 3 determinants.This summarizes our notation for the 2- and 3- permutations.
Let S be the sum of the integer elements of a magic square of order three and let D be the value of the square considered as a determinant. Show that D/S is an integer.
Show that the determinant of the n2 elements in the upper left corner of the Pascal trianglehas the value unity.
List all of the 4-permutations.
Use the determinant to decide if each is singular or nonsingular.
A permutation, regarded as a function from the set {1,..., n} to itself, is one-to-one and onto. Therefore, each permutation has an inverse.(a) Find the inverse of each 2-permutation.(b) Find the inverse of each 3-permutation.This summarizes our notation for the 2- and 3- permutations.
Prove that f is multilinear if and only if for all , ˆˆ V and k1, k2 ˆˆ R, this holds.
Verify the second and third statements in Corollary 3.16.
(a) Show that there are 120 terms in the permutation expansion formula of a 5 × 5 matrix. (b) How many are sure to be zero if the 1, 2 entry is zero?
Show that the inverse of a permutation matrix is its transpose.
Compute the determinant by using the permutation expansion.This summarizes our notation for the 2- and 3- permutations.
Give the permutation expansion of a general 2 × 2 matrix and its transpose.
(a) Find the signum of each 2-permutation.(b) Find the signum of each 3-permutation.
Find the only nonzero term in the permutation expansion of this matrix.Compute that determinant by finding the signum of the associated permutation.
What is the signum of the n-permutation ϕ = (n, n - 1,..., 2, 1)?
Singular or nonsingular? Use the determinant to decide.
Prove these. (a) Every permutation has an inverse. (b) sgn(ϕ-1) = sgn(ϕ) (c) Every permutation is the inverse of another.
Prove that the matrix of the permutation inverse is the transpose of the matrix of the permutation Pϕ-1 = PϕT, for any permutation ϕ.
Show that a permutation matrix with m inversions can be row swapped to the identity in m steps. Contrast this with Corollary 4.5.
For any permutation Ï• let g(Ï•) be the integer defined in this way.(This is the product, over all indices i and j with i (a) Compute the value of g on all 2-permutations. (b) Compute the value of g on all 3-permutations. (c) Prove that g(Ï•) is not 0. (d) Prove this.
Each pair of matrices differ by one row operation. Use this operation to compare det(A) with det(B).(a)(b) (c)
Show this.= (b-a)(c-a)(c-b)
Which real numbers x make this matrix singular?
Do the Gaussian reduction to check the formula for 3 × 3 matrices stated in the preamble to this section.is nonsingular iff aei + bfg + cdh - hfa - idb - gec ‰ 0
Find the volume of this region.
Consider the linear transformation of R3 represented with respect to the standard bases by this matrix.(a) Compute the determinant of the matrix. Does the transformation preserve orientation or reverse it? (b) Find the size of the box defined by these vectors. What is its orientation? (c) Find the
In what way does the definition of a box differ from the definition of a span?
Why doesn't this picture contradict Theorem 1.5?
(a) Suppose that |A| = 3 and that |B| = 2. Find |A2 ∙ BT ∙ B-2 ∙ AT|. (b) Assume that |A| = 0. Prove that |6A3 + 5A2 + 2A| = 0.
Must a transformation t: R2 → R2 that preserves areas also preserve lengths?
Find the area of the triangle in R3 with endpoints (1, 2, 1), (3, -1, 4), and (2, 2, 2). (Area, not volume. The triangle defines a plane-what is the area of the triangle in that plane?)
An alternate proof of Theorem 1.5 uses the definition of determinant functions. (a) Note that the vectors forming S make a linearly dependent set if and only if |S| = 0, and check that the result holds in this case. (b) For the |S| ≠ 0 case, to show that |TS|/|S| = |T| for all transformations,
Give a non-identity matrix with the property that AT = A-1. Show that if AT = A-1 then |A| = ±1. Does the converse hold?
We say that matrices H and G are similar if there is a nonsingular matrix P such that H = P-1GP. Show that similar matrices have the same determinant.
We usually represent vectors in R2 with respect to the standard basis so vectors in the first quadrant have both coordinates positive.Moving counterclockwise around the origin, we cycle thru four regions: Using this basis gives the same counterclockwise cycle. We say these two bases have the same
This question uses material from the optional Determinant Functions Exist subsection. Prove Theorem 1.5 by using the permutation expansion formula for the determinant.
(a) Show that this gives the equation of a line in R2 thru (x2, y2) and (x3, y3).(b) Prove that the area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is (c) Prove that the area of a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3) whose coordinates are integers has an area
Find the cofactor.(a) T2,3 (b) T3,2 (c) T1,3
Find the determinant by expanding(a) On the first row (b) On the second row (c) On the third column.
Find the adjoint of the matrix in Example 1.6.
Find the matrix adjoint to each.(a)(b) (c) (d)
Find the inverse of each matrix in the prior question with Theorem 1.9.(a)(b) (c) (d)
Find the matrix adjoint to this one.
Expand across the first row to derive the formula for the determinant of a 2 × 2 matrix.
Expand across the first row to derive the formula for the determinant of a 3 × 3 matrix.
(a) Give a formula for the adjoint of a 2 × 2 matrix. (b) Use it to derive the formula for the inverse.
Can we compute a determinant by expanding down the diagonal?
Give a formula for the adjoint of a diagonal matrix.
Prove that the transpose of the adjoint is the adjoint of the transpose.
Prove or disprove: adj(adj(T)) = T.
A square matrix is upper triangular if each i, j entry is zero in the part above the diagonal, that is, when i > j. (a) Must the adjoint of an upper triangular matrix be upper triangular? Lower triangular? (b) Prove that the inverse of a upper triangular matrix is upper triangular, if an inverse
This question requires material from the optional Determinants Exist subsection. Prove Theorem 1.5 by using the permutation expansion.
Prove that the determinant of a matrix equals the determinant of its transpose using Laplace's expansion and induction on the size of the matrix.
Show thatwhere Fn is the nth term of 1, 1, 2, 3, 5, ... , x, y, x + y, ... , the Fibonacci sequence, and the determinant is of order n - 1.
Use Cramer's Rule to solve each for each of the variables. (a) x - y = 4 -x + 2y = -7 (b) -2x + y = -2 x - 2y = -2
Prove Cramer's Rule.
Here is an alternative proof of Cramer's Rule that doesn't overtly contain any geometry. Write Xi for the identity matrix with column i replaced by the vector of unknowns x1, . . . , xn. (a) Observe that AXi = Bi. (b) Take the determinant of both sides.
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