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mathematics
linear algebra

Linear Algebra 1st Edition Jim Hefferon - Solutions

Describe the angle between two vectors in R1.
Give a simple necessary and sufficient condition to determine whether the angle between two vectors is acute, right, or obtuse.
Generalize to Rn the converse of the Pythagorean Theorem, that if are perpendicular then
Show that if and only if are perpendicular. Give an example in R2.
Show that if a vector is perpendicular to each of two others then it is perpendicular to each vector in the plane they generate. (Remark. They could generate a degenerate plane-a line or a point-but the statement remains true.)
bisects the angle between them. Illustrate in R2.
Verify that the definition of angle is dimensionally correct: (1) if k > 0 then the cosine of the angle between equals the cosine of the angle between and (2) if k < 0 then the cosine of the angle between is the negative of the cosine of the angle between
Show that the inner product operation is linear: for and k,m ∈ R,
The geometric mean of two positive reals x; y is √ xy. It is analogous to the arithmetic mean (x + y)/2. Use the Cauchy-Schwarz inequality to show that the geometric mean of any x, y ∈ R is less than or equal to the arithmetic mean.
(a) Describe the plane through (1, 1, 5,-1), (2, 2, 2, 0), and (3, 1, 0, 4). (b) Is the origin in that plane?
Astrologers claim to be able to recognize trends in personality and fortune that depend on an individual's birthday by somehow incorporating where the stars were 2000 years ago. Suppose that instead of star-gazers coming up with stuff, math teachers who like linear algebra (we'll call them
A ship is sailing with speed and direction the wind blows apparently (judging by the vane on the mast) in the direction of a vector on changing the direction and speed of the ship from the apparent wind is in the direction of a vector ~b. Find the vector velocity of the wind
Verify the Cauchy-Schwarz inequality by first proving Lagrange's identity:and then noting that the final term is positive. This result is an improvement over Cauchy-Schwarz because it gives a formula for the difference between the two sides. Interpret that difference in R2.
Describe the plane that contains this point and line.
Intersect these planes.
Intersect each pair, if possible(a)(b)
When a plane does not pass through the origin, performing operations on vectors whose bodies lie in it is more complicated than when the plane passes through the origin. Consider the picture in this subsection of the planeand the three vectors with endpoints (2, 0, 0), (1.5, 1, 0), and (1.5, 0,
Show that the line segmentshave the same lengths and slopes if b1 - a1 = d1 - c1 and b2 - a2 = d2 - c2. Is that only if?
Find the reduced echelon form of each matrix.(a)(b)(c)(d)
Get the reduced echelon form of each.(a)(b)
Find each solution set by using Gauss-Jordan reduction and then reading off the parametrization. (a) 2x + y - z = 1 4x - y = 3 (b) x - z = 1 y + 2z - w = 3 x + 2y + 3z - w = 7 (c) x - y + z = 0 y + w = 0 3x - 2y + 3z + w = 0 -y - w = 0 (d) a + 2b + 3c + d - e = 1 3a - b + c + d + e = 3
Give two distinct echelon form versions of this matrix.
List the reduced echelon forms possible for each size. (a) 2×2 (b) 2×3 (c) 3×2 (d) 3×3
What results from applying Gauss-Jordan reduction to a nonsingular matrix?
Consider the following relationship on the set of 2×2 matrices: we say that A is sum-what like B if the sum of all of the entries in A is the same as the sum of all the entries in B. For instance, the zero matrix would be sum-what like the matrix whose first row had two sevens, and whose second
The proof of Lemma 1.5 contains a reference to the i ≠ j condition on the row combination operation. (a) Write down a 2×2 matrix with nonzero entries, and show that the -1 ∙ p1 +p1 operation is not reversed by 1 ∙ p1 + p1. (b) Expand the proof of that lemma to make explicit exactly where it
Consider the set of students in a class. Which of the following relationships are equivalence relations? Explain each answer in at least a sentence. (a) Two students x, y are related if x has taken at least as many math classes as y. (b) Students x, y are related if they have names that start with
Show that each of these is an equivalence on the set of 2×2 matrices. Describe the equivalence classes. (a) Two matrices are related if they have the same product down the diagonal, that is, if the product of the entries in the upper left and lower right are equal. (b) Two matrices are related if
Show that each is not an equivalence on the set of 2×2 matrices. (a) Two matrices A; B are related if a1,1 = -b1,1. (b) Two matrices are related if the sum of their entries are within 5, that is, A is related to B if [(a1;1 + ∙ ∙ ∙ + a2,2) - (b1,1 + ∙ ∙ ∙ + b2,2)] < 5.
The Linear Combination Lemma says which equations can be gotten from Gaussian reduction of a given linear system. (1) Produce an equation not implied by this system. 3x + 4y = 8 2x + y = 3 (2) Can any equation be derived from an inconsistent system?
Describe all matrices in the row equivalence class of these.(a)(b)(c)
In this matrixthe first and second columns add to the third.(a) Show that remains true under any row operation.(b) Make a conjecture.(c) Prove that it holds.
How many row equivalence classes are there?
How big are the row equivalence classes? (a) Show that for any matrix of all zeros, the class is finite. (b) Do any other classes contain only finitely many members?
Give two reduced echelon form matrices that have their leading entries in the same columns, but that are not row equivalent.
Show that any two n × n nonsingular matrices are row equivalent. Are any two singular matrices row equivalent?
Describe all of the row equivalence classes containing these.(a) 2×2 matrices (b) 2×3 matrices (c) 3×2 matrices(d) 3×3 matrices
(a) Show that a vector is a linear combination of members of the set if and only if there is a linear relationship where c0 is not zero.(b) Use that to simplify the proof of Lemma 2.5.
Three truck drivers went into a roadside cafe. One truck driver purchased four sandwiches, a cup of coffee, and ten doughnuts for $8.45. Another driver purchased three sandwiches, a cup of coffee, and seven doughnuts for $6.30. What did the third truck driver pay for a sandwich, a cup of coffee,
Decide if the matrices are row equivalent.(a)(b)(c)(d)(e)
Describe the matrices in each of the classes represented in Example 2.9.
Use Gauss-Jordan reduction to solve each system. (a) x + y = 2 x - y = 0 (b) x - z = 4 2x + 2y = 1 (c) 3x - 2y = 1 6x + y = 1=2 (d) 2x - y = -1 x + 3y - z = 5 y + 2z = 5
Do Gauss-Jordan reduction (a) x + y - z = 3 2x - y - z = 1 3x + y + 2z = 0 (b) x + y + 2z = 0 2x - y + z = 1 4x + y + 5z = 1
Use the computer to solve the two problems that opened this chapter. (a) This is the Statics problem. 40h + 15c = 100 25c = 50 + 50h (b) This is the Chemistry problem. 7h = 7j 8h + 1i = 5j + 2k 1i = 3j 3i = 6j + 1k
A Wheatstone bridge is used to measure resistance.Show that in this circuit if the current flowing through rg is zero then r4 = r2r3=r1. (To operate the device, put the unknown resistance at r4. At rg is a meter that shows the current. We vary the three resistances r1, r2, and r3-typically they
Consider this traffic circle.This is the traffic volume, in units of cars per five minutesWe can set up equations to model how the traffic flows.(a) Adapt Kirchhoff's Current Law to this circumstance. Is it a reasonable modeling assumption?(b) Label the three between-road arcs in the circle with a
This is a network of streetsWe can observe the hourly flow of cars into this network's entrances, and out of its exits.Once inside the network, the traffic may flow in different ways, perhaps filling Willow and leaving Jay mostly empty, or perhaps flowing in some other way. Kirchhoff's Laws give
Calculate the amperages in each part of each network.(a) This is a simple network(b) Compare this one with the parallel case discussed above.(c) This is a reasonably complicated network
Use the computer to solve these systems from the first subsection, or conclude 'many solutions' or 'no solutions'. (a) 2x + 2y = 5 x - 4y = 0 (b) -x + y = 1 x + y = 2 (c) x - 3y + z = 1 x + y + 2z = 14 (d) -x - y = 1 -3x - 3y = 2 (e) 4y + z = 20 2x - 2y + z = 0 x + z = 5 x + y - z = 10 (f) 2x + z +
Use the computer to solve these systems from the second subsection. (a) 3x + 6y = 18 x + 2y = 6 (b) x + y = 1 x - y = -1 (c) x1 + x3 = 4 x1 - x2 + 2x3 = 5 4x1 - x2 + 5x3 = 17 (d) 2a + b - c = 2 2a + c = 3 a - b = 0 (e) x + 2y - z = 3 2x + y + w = 4 x - y + z + w = 1 (f) x + z + w = 4 2x + y - w =
What does the computer give for the solution of the general 2×2 system? ax + cy = p bx + dy = q
Ill-conditioning is not only dependent on the matrix of coefficients. This example [Hamming] shows that it can arise from an interaction between the left and right sides of the system. Let ε be a small real. 3x + 2y + z = 6 2x + 2εy + 2εz = 2 + 4ε x + 2εy - εz = 1 + ε (a) Solve the system by
Consider this system ([Rice]). 0.0003x + 1.556y = 1.569 0.3454x - 2.346y = 1.018 (a) Solve it. (b) Solve it by rounding at each step to four digits.
Rounding inside the computer often has an effect on the result. Assume that your machine has eight significant digits. (a) Show that the machine will compute (2/3) + ((2/3) - (1/3)) as unequal to ((2/3) + (2/3)) - (1/3). Thus, computer arithmetic is not associative. (b) Compare the computer's
This intersect-the-lines problem contrasts with the example discussed above.Illustrate that in this system some small change in the numbers will produce only a small change in the solution by changing the constant in the bottom equation to 1.008 and solving. Compare it to the solution of the
In the first network that we analyzed, with the three resistors in series, we just added to get that they acted together like a single resistor of 10 ohms. We can do a similar thing for parallel circuits. In the second circuit analyzed,the electric current through the battery is 25/6 amperes. Thus,
Name the zero vector for each of these vector spaces. (a) The space of degree three polynomials under the natural operations. (b) The space of 2×4 matrices. (c) The space {f: [0,,1] → R | f is continuous}. (d) The space of real-valued functions of one natural number variable.
Find the additive inverse, in the vector space, of the vector.(a) In P3, the vector -3 - 2x + x2.(b) In the space 2 Ã— 2,(c) In {aex + be-x | a, b ˆˆ R}, the space of functions of the real variable x under the natural operations, the vector 3ex - 2e-x.
For each, list three elements and then show it is a vector space. (a) The set of linear polynomials P1 = {a0 + a1x | a0, a1 ∈ R} under the usual polynomial addition and scalar multiplication operations. (b) The set of linear polynomials {a0 + a1x | a0 - 2a1 = 0}, under the usual polynomial
For each, list three elements and then show it is a vector space. (a) The set of 2 × 2 matrices with real entries under the usual matrix operations. (b) The set of 2 × 2 matrices with real entries where the 2, 1 entry is zero, under the usual matrix operations.
For each, list three elements and then show it is a vector space.(a) The set of three-component row vectors with their usual operations.(b) The setunder the operations inherited from R4.
Show that each of these is not a vector space. (Check closure by listing two members of each set and trying some operations on them.)(a) Under the operations inherited from R3, this set(b) Under the operations inherited from R3, this set(c) Under the usual matrix operations,(d) Under the usual
Define addition and scalar multiplication operations to make the complex numbers a vector space over R.
Which of these are members of the span [{cos2 x, sin2 x}] in the vector space of real-valued functions of one real variable? (a) f(x) = 1 (b) f(x) = 3 + x2 (c) f(x) = sin x (d) f(x) = cos(2x)
Show that the set of linear combinations of the variables x, y, z is a vector space under the natural addition and scalar multiplication operations.
Parametrize each subspace's description. Then express each subspace as a span.(a) The subset {(a b c) | a - c = 0} of the three-wide row vectors(b) This subset of M2Ã—2(c) This subset of M2Ã—2(d) The subset {a + bx + cx3 | a - 2b + c = 0} of P3(e) The subset of P2 of quadratic polynomials p
Prove or disprove that R3 is a vector space under these operations.(a)(b)
For each, decide if it is a vector space, the intended operations are the natural ones.(a) The diagonal 2 Ã— 2 matrices(b) This set of 2 Ã— 2 matrices(c) This set(d) The set of functions {f : R †’ R | df/dx + 2f = 0}(e) The set of functions {f : R †’ R | df/dx + 2f = 1}
Prove or disprove that this is a vector space: the real-valued functions f of one real variable such that f(7) = 0.
Show that the set R+ of positive reals is a vector space when we interpret 'x+y' to mean the product of x and y (so that 2 + 3 is 6), and we interpret 'r ∙ x' as the r-th power of x.
Is {(x, y) | x, y ∈ R} a vector space under these operations? (a) (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2) and r ∙ (x, y) = (rx, y) (b) (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2) and r ∙ (x, y) = (rx, 0)
Give an example of each or explain why it would be impossible to do so. (a) A nonempty subset of M2×2 that is not a subspace. (b) A set of two vectors in R2 that does not span the space.
At this point "the same" is only an intuition, but nonetheless for each vector space identify the k for which the space is "the same" as Rk.(a) The 2 Ã— 3 matrices under the usual operations(b) The n Ã— m matrices (under their usual operations)(c) This set of 2 Ã— 2 matrices(d) This set of
Usingto represent vector addition and for scalar multiplication, restate the definition of vector space.
Prove these.(a) For any is an additive inverse of then is an additive inverse of. So a vector is an additive inverse of any additive inverse of itself.(b) Vector addition left-cancels: if implies that
Show that for any subset S of a vector space, the span of the span equals the span [[S]] = [S]. (Members of [S] are linear combinations of members of S. Members of [[S]] are linear combinations of linear combinations of members of S.)
All of the subspaces that we've seen in some way use zero in their description. For example, the subspace in Example 2.3 consists of all the vectors from R2 with a second component of zero. In contrast, the collection of vectors from R2 with a second component of one does not form a subspace (it is
In a vector space every element has an additive inverse. Can some elements have two or more?
(a) Prove that every point, line, or plane thru the origin in R3 is a vector space under the inherited operations.(b) What if it doesn't contain the origin?
Using the idea of a vector space we can easily reprove that the solution set of a homogeneous linear system has either one element or infinitely many elements. Assume that (a) Prove that if and only if r = 0.(b) Prove that if and only if r1 = r2.(c) Prove that any nontrivial vector space is
Is this a vector space under the natural operations: the real-valued functions of one real variable that are differentiable?
A vector space over the complex numbers C has the same definition as a vector space over the reals except that scalars are drawn from C instead of from R. Show that each of these is a vector space over the complex numbers. (Recall how complex numbers add and multiply: (a0 + a1i) + (b0 + b1i) = (a0
Show that if a vector is in the span of a set then adding that vector to the set won't make the span any bigger. Is that also 'only if'?
(a) Prove that for any four vectors we can associate their sum in any way without changing the result.This allows us to write without ambiguity.(b) Prove that any two ways of associating a sum of any number of vectors give the same sum. (Use induction on the number of vectors.)
Example 1.5 gives a subset of R2 that is not a vector space, under the obvious operations, because while it is closed under addition, it is not closed under scalar multiplication. Consider the set of vectors in the plane whose components have the same sign or are 0. Show that this set is closed
Is the relation 'is a subspace of' transitive? That is, if V is a subspace of W and W is a subspace of X, must V be a subspace of X?
Because 'span of' is an operation on sets we naturally consider how it interacts with the usual set operations.(a) If S ⊂ T are subsets of a vector space, is [S] ⊂ [T]? Always? Sometimes? Never?(b) If S, T are subsets of a vector space, is [S ∪ T] = [S] ∪ [T]?(c) If S, T are subsets of a
Reprove Lemma 2.15 without doing the empty set separately.
Find a structure that is closed under linear combinations, and yet is not a vector space.
Is the vector in the span of the set?
Decide if the vector lies in the span of the set, inside of the space.(a)(b) x - x3, {x2, 2x + x2, x + x3}, in P3(c)
Which of these sets spans R3? That is, which of these sets has the property that any three-tall vector can be expressed as a suitable linear combination of the set's elements?(a)(b)(c)(d)(e)
Find a set to span the given subspace of the given space.(a) The xz-plane in R3(b)(c)(d) {a0 + a1x + a2x2 + a3x3 | a0 + a1 = 0 and a2 - a3 = 0} in P3(e) The set P4 in the space P4(f) M2×2 in M2×2
Is R2 a subspace of R3?
Decide if each is a subspace of the vector space of real-valued functions of one real variable. (a) The even functions {f : R → R | f(-x) = f(x) for all x}. For example, two members of this set are f1(x) = x2 and f2(x) = cos(x). (b) The odd functions {f : R → R | f(-x) = -f(x) for all x}. Two
Example 2.16 says that for any vector that is an element of a vector space V, the set {r ∙ | r ∈ R} is a subspace of V. (This is of course, simply the span of the singleton set {}.) Must any such subspace be a proper subspace, or can it be improper?
Example 2.19 shows that R3 has infinitely many subspaces. Does every nontrivial space have infinitely many subspaces?
Finish the proof of Lemma 2.9.
Show that each vector space has only one trivial subspace.
(a) What is the difference between this sum of three vectors and the sum of the first two of these three?(b) What is the difference between the prior sum and the sum of just the first one vector?(c) What should be the difference between the prior sum of one vector and the sum of no vectors?(d) So

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