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mathematics
linear algebra

Linear Algebra 1st Edition Jim Hefferon - Solutions

What is the minimal polynomial of a diagonal matrix?
A projection is any transformation t such that t2 = t. (For instance, consider the transformation of the plane R2 projecting each vector onto its first coordinate. If we project twice then we get the same result as if we project just once.) What is the minimal polynomial of a projection?
The first two items of this question are review.(a) Prove that the composition of one-to-one maps is one-to-one.(b) Prove that if a linear map is not one-to-one then at least one nonzero vector from the domain maps to the zero vector in the codomain.(c) Verify the statement, excerpted here, that
True or false: for a transformation on an n dimensional space, if the minimal polynomial has degree n then the map is diagonalizable.
Let f(x) be a polynomial. Prove that if A and B are similar matrices then f(A) is similar to f(B).(a) Now show that similar matrices have the same characteristic polynomial.(b) Show that similar matrices have the same minimal polynomial.(c) Decide if these are similar.
(a) Show that a matrix is invertible if and only if the constant term in its minimal polynomial is not 0. (b) Show that if a square matrix T is not invertible then there is a nonzero matrix S such that ST and TS both equal the zero matrix.
(a) Finish the proof of Lemma 1.7.(b) Give an example to show that the result does not hold if t is not linear.
Any transformation or square matrix has a minimal polynomial. Does the converse hold?
Prove or disprove: two n × n matrices are similar if and only if they have the same characteristic and minimal polynomials.
The trace of a square matrix is the sum of its diagonal entries.(a) Find the formula for the characteristic polynomial of a 2 × 2 matrix.(b) Show that trace is invariant under similarity, and so we can sensibly speak of the 'trace of a map'. (Hint: see the prior item.)(c) Is trace invariant under
To use Definition 2.7 to check whether a subspace is t invariant, we seemingly have to check all of the infinitely many vectors in a (nontrivial) subspace to see if they satisfy the condition. Prove that a subspace is t invariant if and only if its sub basis has the property that for all of its
Is t invariance preserved under intersection? Under union? Complementation? Sums of subspaces?
Do the check for Example 2.4.
Let Pj(R) be the vector space over the reals of degree j polynomials. Show that if j ≤ k then Pj(R) is an invariant subspace of Pk(R) under the differentiation operator. In P7(R), does any of P0(R), . . . , P6(R) have an invariant complement?
In Pn(R), the vector space (over the reals) of degree n polynomials, ∈ = {p(x) ∈ Pn(R) j p( - x) = p(x) for all x} and O = {p(x) ∈ Pn(R) | {(- x) = - p(x) for all x} are the even and the odd polynomials; p(x) = x2 is even while p(x) = x3 is odd. Show that they are subspaces. Are they
Lemma 2.9 says that if M and N are invariant complements then t has a representation in the given block form (with respect to the same ending as starting basis, of course). Does the implication reverse?
A matrix S is the square root of another T if S2 = T. Show that any nonsingular matrix has a square root.
Each matrix is in Jordan form. State its characteristic polynomial and its minimal polynomial.(a)(b)(c)(d)(e)(f)(g)(h)
Find the Jordan form and a Jordan basis for each matrix(a)(b)(c)(d)(e)(f)(g)
Find all possible Jordan forms of a transformation with characteristic polynomial (x - 1)2(x + 2)2.
Find all possible Jordan forms of a transformation with characteristic polynomial (x - 1)3(x + 2).
Find all possible Jordan forms of a transformation with characteristic polynomial (x - 2)3(x + 1) and minimal polynomial (x - 2)2(x + 1).
Diagonalize these.(a)(b)
Find the Jordan matrix representing the differentiation operator on P3.
Decide if these two are similar.
Find the Jordan form of this matrix.Also give a Jordan basis.
How many similarity classes are there for 3 × 3 matrices whose only eigenvalues are -3 and 4?
Prove that a matrix is diagonalizable if and only if its minimal polynomial has only linear factors.
Use ten iterations to estimate the largest eigenvalue of these matrices, starting from the vector with components 1 and 2. Compare the answer with the oneobtained by solving the characteristic equation.(a)(b)
Redo the prior exercise by iterating until |→vk| - |→vk-1| has absolute value less than 0:01 At each step, normalize by dividing each vector by its length. How many iterations does it take? Are the answers significantly different?
Use ten iterations to estimate the largest eigenvalue of these matrices, starting from the vector with components 1, 2, and 3. Compare the answer with the one obtained by solving the characteristic equation.(a)(b)
Redo the prior exercise by iterating until |→vk| - |→vk-1j has absolute value less than 0:01. At each step, normalize by dividing each vector by its length. How many iterations does it take? Are the answers significantly different?
What happens if c1 = 0? That is, what happens if the initial vector does not to have any component in the direction of the relevant eigenvector?
How can we adapt the method of powers to find the smallest eigenvalue?
For the park discussed above, what should be the initial park population in the case where the populations decline by 11% every year?
What will happen to the population of the park in the event of a growth in world population of 1% per year? Will it lag the world growth, or lead it? Assume that the initial park population is ten thousand, and the world population is one hundred thousand, and calculate over a ten year span.
The park discussed above is partially fenced so that now, every year, only 5% of the animals from inside of the park leave (still, about 1% of the animals from the outside find their way in). Under what conditions can the park maintain a stable population now?
Suppose that a species of bird only lives in Canada, the United States, or in Mexico. Every year, 4% of the Canadian birds travel to the US, and 1% of them travel to Mexico. Every year, 6% of the US birds travel to Canada, and 4% go to Mexico. From Mexico, every year 10% travel to the US, and 0% go
A square matrix is stochastic if the sum of the entries in each column is one. The Google matrix is computed by taking a combination G = α * H + (1 - α) * S of two stochastic matrices. Show that G must be stochastic.
For this web of pages, the importance of each page should be equal. Verify it for Î± = 0:85.
Give the importance ranking for this web of pages.(a) Use Î± = 0:85. (b) Use Î± = 0:95. (c) Observe that while p3 is linked-to from all other pages, and therefore seems important, it is not the highest ranked page. What is the highest ranked page? Explain.
How many months until the number of Fibonacci rabbit pairs passes a thousand? Ten thousand? A million?
Solve each homogeneous linear recurrence relations.(a) f(n) = 5f(n - 1) - 6f(n - 2)(b) f(n) = 4f(n - 2)(c) f(n) = 5f(n - 1) - 2f(n - 2) - 8f(n - 3)
Give a formula for the relations of the prior exercise, with these initial conditions. (a) f(0) = 1, f(1) = 1 (b) f(0) = 0, f(1) = 1 (c) f(0) = 1, f(1) = 1, f(2) = 3.
Check that the isomorphism given between S and Rk is a linear map.
Show that the characteristic equation of the matrix is as stated, that is, is the polynomial associated with the relation. (Hint: expanding down the final column and using induction will work.)
(This refers to the value T(64) = 18; 446; 744; 073; 709; 551; 615 given in the computer code below.) Transferring one disk per second, how many years would it take the priests at the Tower of Hanoi to finish the job?
A three-digit number has two properties. The tens-digit and the ones-digit add up to 5. If the number is written with the digits in the reverse order, and then subtracted from the original number, the result is 792. Use a system of equations to find all of the three-digit numbers with these
Find all of the six-digit numbers in which the first digit is one less than the second, the third digit is half the second, the fourth digit is three times the third and the last two digits form a number that equals the sum of the fourth and fifth. The sum of all the digits is 24. (From The MENSA
Each sentence below has at least two meanings. Identify the source of the double meaning, and rewrite the sentence (at least twice) to clearly convey each meaning. 1. They are baking potatoes. 2. He bought many ripe pears and apricots. 3. She likes his sculpture. 4. I decided on the bus.
The following sentence, due to Noam Chomsky, has a correct grammatical structure, but is meaningless. Critique its faults. "Colorless green ideas sleep furiously."
Read the following sentence and form a mental picture of the situation. The baby cried and the mother picked it up. What assumptions did you make about the situation?
This problem appears in a middle-school mathematics textbook: Together Dan and Diane have $20. Together Diane and Donna have $15. How much do the three of them have in total? (Transition Mathematics, Second Edition, Scott Foresman Addison Wesley, 1998. Problem 5(1.19.)
In Example TMP two different prices were considered for marketing standard mix with the revised recipes (one-third peanuts in each recipe). Selling standard mix at $5.50 resulted in selling the minimum amount of the fancy mix and no bulk mix. At $5.25 it was best for profits to sell the maximum
We have seen in this section that systems of linear equations have limited possibilities for solution sets, and we will shortly prove Theorem PSSLS that describes these possibilities exactly. This exercise will show that if we relax the requirement that our equations be linear, then the
Proof Technique D asks you to formulate a definition of what it means for a whole number to be odd. What is your definition? (Do not say "the opposite of even.") Is 6 odd? Is 11 odd? Justify your answers by using your definition.
Explain why the second equation operation in Definition EO requires that the scalar be nonzero, while in the third equation operation this restriction on the scalar is not present.
Find all solutions to the system of linear equations. Use your favorite computing device to row-reduce the augmented matrices for the systems, and write the solutions as a set, using correct set notation. 1. 2x1 - 3x2 + x3 + 7x4 = 14 2x1 + 8x2 - 4x3 + 5x4 = - 1 x1 + 3x2 - 3x3 = 4 - 5x1 + 2x2 + 3x3
Row-reduce the matrix without the aid of a calculator, indicating the row operations you are using at each step using the notation of Definition RO.1.2.
Consider the two 3 Ã— 4 matrices below1. Row-reduce each matrix and determine that the reduced row-echelon forms of B and C are identical. From this argue that B and C are row-equivalent. 2. In the proof of Theorem RREFU, we begin by arguing that entries of row-equivalent matrices are
You keep a number of lizards, mice and peacocks as pets. There are a total of 108 legs and 30 tails in your menagerie. You have twice as many mice as lizards. How many of each creature do you have?
A parking lot has 66 vehicles (cars, trucks, motorcycles and bicycles) in it. There are four times as many cars as trucks. The total number of tires (4 per car or truck, 2 per motorcycle or bicycle) is 252. How many cars are there? How many bicycles?
Prove that each of the three row operations (Definition RO) is reversible. More precisely, if the matrix B is obtained from A by application of a single row operation, show that there is a single row operation that will transform B back into A.
Find the solution set of the system of linear equations. Give the values of n and r, and interpret your answers in light of the theorems of this section. 1. x1 + 4x2 + 3x3 - x4 = 5 x1 - x2 + x3 + 2x4 = 6 4x1 + x2 + 6x3 + 5x4 = 9 2. x1 - 2x2 + x3 - x4 = 3 2x1 - 4x2 + x3 + x4 = 2 x1 - 2x2 - 2x3 + 3x4
The details for Archetype J include several sample solutions. Verify that one of these solutions is correct (any one, but just one). Based only on this evidence, and especially without doing any row operations, explain how you know this system of linear equations has infinitely many solutions.
Say as much as possible about each system's solution set. Be sure to make it clear which theorems you are using to reach your conclusions. 1. A consistent system of 8 equations in 6 variables. 2. A consistent system of 6 equations in 8 variables.
An inconsistent system may have r > n. If we try (incorrectly!) to apply Theorem FVCS to such a system, how many free variables would we discover?
Suppose that the coefficient matrix of a consistent system of linear equations has two columns that are identical. Prove that the system has infinitely many solutions.
Consider the system of linear equations LS(A, b), and suppose that every element of the vector of constants b is a common multiple of the corresponding element of a certain column of A. More precisely, there is a complex number and a column index j, such that [b]i = α [A]ij for all i. Prove that
Solve the given homogeneous linear system. Compare your results to the results of the corresponding exercise in Section TSS. 1. x1 + 4x2 + 3x3 - x4 = 0 x1 - x2 + x3 + 2x4 = 0 4x1 + x2 + 6x3 + 5x4 = 0 2. x1 - 2x2 + x3 - x4 = 0 2x1 - 4x2 + x3 + x4 = 0 x1 - 2x2 - 2x3 + 3x4 = 0
Solve the given homogeneous linear system. Compare your results to the results of the corresponding exercise in Section TSS. 1. x1 + 2x2 + 3x3 = 0 2x1 - x2 + x3 = 0 3x1 + x2 + x3 = 0 x2 + 2x3 = 0 2. x1 + 2x2 + 3x3 = 0 2x1 - x2 + x3 = 0 3x1 + x2 + x3 = 0 5x2 + 2x3 = 0
Compute the null space of the matrix A, N(A).
Find the null space of the matrix B, N(B).
Find all solutions to the linear system: x + y = 5 2x - y = 3
Say as much as possible about each system's solution set. Be sure to make it clear which theorems you are using to reach your conclusions. 1. A homogeneous system of 8 equations in 8 variables. 2. A homogeneous system of 8 equations in 9 variables.
Prove or disprove the given statement. "A system of linear equations is homogeneous if and only if the system has the zero vector as a solution"
Suppose that two systems of linear equations are equivalent. Prove that if the first system is homogeneous, then the second system is homogeneous. Notice that this will allow us to conclude that two equivalent systems are either both homogeneous or both not homogeneous.
Consider the homogeneous system of linear equations LS(A, 0), and suppose thatis one solution to the system of equations. Prove that is also a solution to LS(A, 0).
Find the null space of the matrix E below.
Let A be the coefficient matrix of the system of equations below. Is A nonsingular or singular? Explain what you could infer about the solution set for the system based only on what you have learned about A being singular or nonsingular. - x1 + 5x2 = - 8 - 2x1 + 5x2 + 5x3 + 2x4 = 9 - 3x1 - x2 + 3x3
Say as much as possible about each system's solution set. Be sure to make it clear which theorems you are using to reach your conclusions. 1. 6 equations in 6 variables, singular coefficient matrix. 2. A system with a nonsingular coefficient matrix, not homogeneous.
Suppose that A is a square matrix, and B is a matrix in reduced row-echelon form that is row-equivalent to A. Prove that if A is singular, then the last row of B is a zero row.
Suppose that A is a nonsingular matrix and A is row-equivalent to the matrix B. Prove that B is nonsingular.
Suppose that A is a square matrix of size n × n and that we know there is a single vector b ∈ Cn such that the system LS(A, b) has a unique solution. Prove that A is a nonsingular matrix.
Provide an alternative for the second half of the proof of Theorem NMUS, without appealing to properties of the reduced row-echelon form of the coefficient matrix. In other words, prove that if A is nonsingular, then LS(A, b) has a unique solution for every choice of the constant vector b.
Prove Property CC of Theorem VSPCV. Write your proof in the style of the proof of Property DSAC given in this section.
Consider each archetype that is a system of equations. Write elements of the solution set in vector form, as guaranteed by Theorem VFSLS. Archetype A, Archetype B, Archetype C, Archetype D, Archetype E, Archetype F, Archetype G, Archetype H, Archetype I, Archetype J
Find the vector form of the solutions to the system of equations below. 2x1 - 4x2 + 3x3 + x5 = 6 x1 - 2x2 - 2x3 + 14x4 - 4x5 = 15 x1 - 2x2 + x3 + 2x4 + x5 = -1 -2x1 + 4x2 - 12x4 + x5 = -7
Find the vector form of the solutions to the system of equations below. -2x1 - 1x2 - 8x3 + 8x4 + 4x5 - 9x6 - 1x7 - 1x8 - 18x9 = 3 3x1 - 2x2 + 5x3 + 2x4 - 2x5 - 5x6 + 1x7 + 2x8 + 15x9 = 10 4x1 - 2x2 + 8x3 + 2x5 - 14x6 - 2x8 + 2x9 = 36 -1x1 + 2x2 + 1x3 - 6x4 + 7x6 - 1x7 - 3x9 = -8 3x1 + 2x2 + 13x3 -
Solve the given vector equation for x, or explain why no solution exists:
Example TLC asks if the vectorcan be written as a linear combination of the four vectors Can it? Can any vector in C6 be written as a linear combination of the four vectors u1, u2, u3, u4?
At the end of Example VFS, the vector w is claimed to be a solution to the linear system under discussion. Verify that w really is a solution. Then determine the four scalars that express w as a linear combination of c, u1, u2, u3.
For each archetype that is a system of equations, consider the corresponding homogeneous system of equations. Write elements of the solution set to these homogeneous systems in vector form, as guaranteed by Theorem VFSLS. Then write the null space of the coefficient matrix of each system as the
Suppose thatLet W = (S) and let Is x ˆˆ W? If so, provide an explicit linear combination that demonstrates this.
Suppose thatLet W = (S) and let Is y ˆˆ W? If so, provide an explicit linear combination that demonstrates this.
SupposeIs in (R)?
Suppose thatLet W = (S) and let Is y ˆˆ W? If so, provide an explicit linear combination that demonstrates this.
Suppose thatLet W = (S) and let Is w ˆˆ W? If so, provide an explicit linear combination that demonstrates this.
Solve the given vector equation for Î±, or explain why no solution exists:
Let A be the matrix below.1. Find a set S so that N(A) = (S).2. Ifthen show directly that z ˆˆ N(A). 3. Write z as a linear combination of the vectors in S.

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