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mathematics
linear algebra

Linear Algebra 1st Edition Jim Hefferon - Solutions

Use Cramer's Rule to give a formula for the solution of a two equations/two unknowns linear system.
Can Cramer's Rule tell the difference between a system with no solutions and one with infinitely many?
The first picture in this Topic (the one that doesn't use determinants) shows a unique solution case. Produce a similar picture for the case of infinitely many solutions, and the case of no solutions.
Compute the determinant of each of these by hand using the two methods discussed above.(a)(b) (c) Count the number of multiplications and divisions used in each case, for each of the methods.
The use by the timing routine of do_matrix has a bug. That routine does two things, generate a random matrix and then do gauss_method on it, and the timing number returned is for the combination. Produce code that times only the gauss_method routine.
What 10 × 10 array can you invent that takes your computer the longest time to reduce? The shortest?
Use ChiÃ²'s Method to find each determinant.(a)(b)
What if a1,1 is zero?
The Rule of Sarrus is a mnemonic that many people learn for the 3 Ã— 3 determinant formula. To the right of the matrix, copy the first two columns.Then the determinant is the sum of the three upper-left to lower-right diagonals minus the three lower-left to upper-right diagonals aei +
Prove Chiò's formula.
What is the equation of this point?
(a) Find the line incident on these points in the projective plane.(b) Find the point incident on both of these projective lines. (1 2 3), (4 5 6)
Find the formula for the line incident on two projective points. Find the formula for the point incident on two projective lines.
Prove that the definition of incidence is independent of the choice of the representatives of p and L. That is, if p1, p2, p3, and q1, q2, q3 are two triples of homogeneous coordinates for p, and L1, L2, L3, and M1, M2, M3 are two triples of homogeneous coordinates for L, prove that p1L1 + p2L2 +
Give a drawing to show that central projection does not preserve circles, that a circle may project to an ellipse. Can a (non-circular) ellipse project to a circle?
Give the formula for the correspondence between the non-equatorial part of the antipodal modal of the projective plane, and the plane z = 1.
Give the formula for the correspondence between the non-equatorial part of the antipodal modal of the projective plane, and the plane z = 1. Discuss.
(Pappus's Theorem) Assume that T0, U0, and V0 are collinear and that T1, U1, and V1 are collinear. Consider these three points: (i) the intersection V2 of the lines T0U1 and T1U0, (ii) the intersection U2 of the lines T0V1 and T1V0, and (iii) the intersection T2 of U0V1 and U1V0. (a) Draw a
Exhibit an nontrivial similarity relationship by letting t : C2 †’ C2 act in this way,picking two bases B,D, and representing t with respect to them, ^T = RepB,B(t) and T = RepD,D(t). Then compute the P and P-1 to change bases from B to D and back again.
Show that the inverse of a diagonal matrix is the diagonal of the the inverses, if no element on that diagonal is zero. What happens when a diagonal entry is zero?
Explain Example 1.4 in terms of maps.
Are there two matrices A and B that are similar while A2 and B2 are not similar?
Find a formula for the powers of this matrix.
Show that similarity is an equivalence relation. (The definition given earlier already reflects this, so instead start here with the definition that ^T is similar to T if ^T = PTP-1.)
Consider a matrix representing, with respect to some B,B, reflection across the x-axis in R2. Consider also a matrix representing, with respect to some D,D, reflection across the y-axis. Must they be similar?
Prove that similarity preserves determinants and rank. Does the converse hold?
Is there a matrix equivalence class with only one matrix similarity class inside? One with infinitely many similarity classes?
Can two different diagonal matrices be in the same similarity class?
Prove that if two matrices are similar then their k-th powers are similar when k > 0. What if k ≤ 0?
Let p(x) be the polynomial cnxn + ∙∙∙∙∙∙ + c1x + c0. Show that if T is similar to S then p(T) = cnTn + ∙∙∙∙∙∙ + c1T + c0I is similar to p(S) = cnSn + ∙∙∙∙∙∙ + c1S + c0I.
List all of the matrix equivalence classes of 1 × 1 matrices. Also list the similarity classes, and describe which similarity classes are contained inside of each matrix equivalence class.
Does similarity preserve sums?
Show that if T - λI and N are similar matrices then T and N + λI are also similar.
Repeat Example 2.5 for the matrix from Example 2.2.
Diagonalize these upper triangular matrices
Give two same-sized diagonal matrices that are not similar. Must any two different diagonal matrices come from different similarity classes?
The equation ending Example 2.5is a bit jarring because for P we must take the first matrix, which is shown as an inverse, and for P-1 we take the inverse of the first matrix, so that the two -1 powers cancel and this matrix is shown without a superscript -1.(a) Check that this nicer-appearing
Show that the P used to diagonalize in Example 2.5 is not unique
We can ask how diagonalization interacts with the matrix operations. Assume that t, s: V → V are each diagonalizable. Is ct diagonalizable for all scalars c? What about t + s? t ○ s?
Show that matrices of this form are not diagonalizable.
Show that each of these is diagonalizable.(a)(b)x,y,z scalars
For each, find the characteristic polynomial and the eigenvalues.
Prove that if a,......, d are all integers and a + b = c + d thenhas integral eigenvalues, namely a + b and a - c.
Prove that if T is nonsingular and has eigenvalues λ1,......,, λn then T-1 has eigenvalues 1 = λ1,...., = λn. Is the converse true?
Suppose that T is n × n and c, d are scalars.(a) Prove that if T has the eigenvalue λ with an associated eigenvector →v then →v is an eigenvector of cT + dI associated with eigenvalue cλ + d.(b) Prove that if T is diagonalizable then so is cT + dI.
Show that λ is an eigenvalue of T if and only if the map represented by T - λI is not an isomorphism.
(a) Show that if λ is an eigenvalue of A then λk is an eigenvalue of Ak. (b) What is wrong with this proof generalizing that? "If λ is an eigenvalue of A and λ is an eigenvalue for B, then λμ is an eigenvalue for AB, for, if A→x = λ→x and B→x = λ→x then AB→x = Aλ→x = λA→x
Do matrix equivalent matrices have the same eigenvalues?
For each matrix, find the characteristic equation, and the eigenvalues and associated eigenvectors.
Find the characteristic equation, and the eigenvalues and associated eigenvectors for this matrix. Hint. The eigenvalues are complex.
Suppose that P is a nonsingular n × n matrix. Show that the similarity transformation map tP : Mn × n → Mn × n sending T → PTP-1 is an isomorphism.
Show that if A is an n square matrix and each row (column) sums to c then c is a characteristic root of A. ("Characteristic root" is a synonym for eigenvalue.)
Find the characteristic polynomial, the eigenvalues, and the associated eigenvectors of this matrix.
For each matrix, find the characteristic equation, and the eigenvalues and associated eigenvectors.
Let t: P2 → P2 be a0 + a1x + a2x2 → (5a0 + 6a1 + 2a2) - (a1 + 8a2)x + (a0 - 2a2)x2∙ Find its eigenvalues and the associated eigenvectors.
Find the eigenvalues and associated eigenvectors of the differentiation operator d/dx: P3→ P3.
Find the formula for the characteristic polynomial of a 2 × 2 matrix.
Example 1.4 shows that the only matrix similar to a zero matrix is itself and that the only matrix similar to the identity is itself.(a) Show that the 1 × 1 matrix whose single entry is 2 is also similar only to itself.(b) Is a matrix of the form cI for some scalar c similar only to itself?(c) Is
Consider this transformation of C3and these bases.We will compute the parts of the arrow diagram to represent the transformation using two similar matrices.(a) Draw the arrow diagram, specialized for this case.(b) Compute T = RepB,B(t).(c) Compute ^T = RepD,D(t).(d) Compute the matrices for other
Consider the transformation t : P2 → P2 described by x2 → x + 1, x → x2 - 1, and 1 → 3.(a) Find T = RepB,B(t) where B = (x2, x,1).(b) Find ^T = RepD.D(t) where D = (1,1 + x,1 + x + x2).(c) Find the matrix P such that ^T = PTP-1.
Let T represent t : C2 †’ C2 with respect to B; B.We will convert to the matrix representing t with respect to D,D.(a) Draw the arrow diagram.(b) Give the matrix that represents the left and right sides of that diagram, in the direction that we traverse the diagram to make the conversion.(c)
For each map, give the chain of range spaces and the chain of null spaces, and the generalized range space and the generalized null space.(a) t0: P2 †’ P2, a + bx + cx2 †” b + cx2(b) t1: R2 †’ R2,(c) t2 : P2 †’ P2, a + bx + cx2 †” b + cx + ax2d) t3: R3 †’ R3,
Check that a subspace must be of dimension less than or equal to the dimension of its superspace. Check that if the subspace is proper (the subspace does not equal the superspace) then the dimension is strictly less. (This is used in the proof of Lemma 1.4.)
Prove that the generalized range space R1(t) is the entire space, and the generalized null space N∞(t) is trivial, if the transformation t is nonsingular. Is his 'only if' also?
Verify the null space half of Lemma 1.4.
Give an example of a transformation on a three dimensional space whose range has dimension two. What is its null space? Iterate your example until the range space and null space stabilize.
Show that the range space and null space of a linear transformation need not be disjoint. Are they ever disjoint?
Give the chains of range spaces and null spaces for the zero and identity transformations.
What is the index of nilpotency of the right-shift operator, here acting on the space of triples of reals? (x, y, z) ↔ (0, x,y)
For each string basis state the index of nilpotency and give the dimension of the range space and null space of each iteration of the nilpotent map.(a)(b)(c)Also give the canonical form of the matrix
Decide which of these matrices are nilpotent.
Find the canonical form of 0this matrix.
Consider the matrix from Example 2.18.(a) Use the action of the map on the string basis to give the canonical form.(b) Find the change of basis matrices that bring the matrix to canonical form.(c) Use the answer in the prior item to check the answer in the first item.
Each of these matrices is nilpotent.
Describe the effect of left or right multiplication by a matrix that is in the canonical form for nilpotent matrices.
Is nilpotence invariant under similarity? That is, must a matrix similar to a nilpotent matrix also be nilpotent? If so, with the same index?
Show that the only eigenvalue of a nilpotent matrix is zero.
Let t: V → V be a linear transformation and suppose →v ∈ V is such that tk(→v) = →0 but tk-1(→v) ↔ →0. Consider the t-string h→v, t(→v),....., tk-1(→v)i.(a) Prove that t is a transformation on the span of the set of vectors in the string, that is, prove that t restricted to
Finish the proof of Theorem 2.15.
Show that the terms 'nilpotent transformation' and 'nilpotent matrix', as given in Definition 2.7, fit with each other: a map is nilpotent if and only if it is represented by a nilpotent matrix. (Is it that a transformation is nilpotent if an only if there is a basis such that the map's
Let T be nilpotent of index four. How big can the range space of T3 be?
Recall that similar matrices have the same eigenvalues. Show that the converse does not hold.
Lemma 2.1 shows that any for any linear transformation t: V → V the restriction t: R∞(t) →R∞(t) is one-to-one. Show that it is also onto, so it is an automorphism. Must it be the identity map?
Prove that a nilpotent matrix is similar to one that is all zeros except for blocks of super-diagonal ones.
Prove that if a transformation has the same range space as null space. Then the dimension of its domain is even.
Prove that if two nilpotent matrices commute then their product and sum are also nilpotent.
What are the possible minimal polynomials if a matrix has the given characteristic polynomial?(a) (x - 3)4 (b) (x + 1)3(x - 4) (c) (x - 2)2(x - 5)2(d) (x + 3)2(x - 1)(x - 2)2What is the degree of each possibility?
Find the minimal polynomial of each matrix.
Find the minimal polynomial of this matrix
What is the minimal polynomial of the differentiation operator d/dx on Pn?
Find the minimal polynomial of matrices of this formwhere the scalar Î» is fixed (i.e., is not a variable).
What is the minimal polynomial of the transformation of Pn that sends p(x) to p(x + 1)?
What is the minimal polynomial of the map π : C3 → C3 projecting onto the first two coordinates?
Find the Jordan form from the given data.(a) The matrix T is 5 × 5 with the single eigenvalue 3. The nullities of the powers are: T - 3I has nullity two, (T - 3I)2 has nullity three, (T - 3I)3 has nullity four, and (T - 3I)4 has nullity five.(b) The matrix S is 5 × 5 with two eigenvalues. For the
Find the change of basis matrices for each example. (a) Example 2.15 (b) Example 2.16 (c) Example 2.17
Verify Lemma 1.9 for 2 × 2 matrices by direct calculation.
Prove that the minimal polynomial of an n × n matrix has degree at most n (not n2 as a person might guess from this subsection's opening). Verify that this maximum, n, can happen.
Show that, on a nontrivial vector space, a linear transformation is nilpotent if and only if its only eigenvalue is zero.
What is the minimal polynomial of a zero map or matrix? Of an identity map or matrix?
Find all possible Jordan forms of a transformation with characteristic polynomial (x - 2)4(x + 1) and minimal polynomial (x - 2)2(x + 1).

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