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linear algebra

Linear Algebra 1st Edition Jim Hefferon - Solutions

Let V = W1 + ⊕∙ ∙ ∙ ∙⊕ Wk and for each index i suppose that Si is a linearly independent subset of Wi. Prove that the union of the Si's is linearly independent.
A matrix is symmetric if for each pair of indices i and j, the i, j entry equals the j, i entry. A matrix is antisymmetric if each i, j entry is the negative of the j, I entry. (a) Give a symmetric 2 × 2 matrix and an antisymmetric 2 × 2 matrix. (Remark. For the second one, be careful about the
Let W1,W2,W3 be subspaces of a vector space. Prove that (W1 ∩ W2) + (W1 ∩ W3) ⊂ W1 ∩ (W2 + W3). Does the inclusion reverse?
The example of the x-axis and the y-axis in R2 shows that W1 ⊕ W2 = V does not imply that W1 ∪ W2 = V. Can W1 ⊕ W2 = V and W1 ∪ W2 = V happen?
Consider Corollary 4.13. Does it work both ways-that is, supposing that V = W1 +∙ ∙ ∙ ∙+Wk, is V = W1 ⊕∙ ∙ ∙⊕ Wk if and only if dim(V) = dim(W1)+∙ ∙ ∙ ∙+dim(Wk)?
We can ask about the algebra of the '+' operation. (a) Is it commutative, is W1 +W2 = W2 +W1? (b) Is it associative, is (W1 +W2) +W3 = W1 + (W2 +W3)? (c) Let W be a subspace of some vector space. Show that W +W = W. (d) Must there be an identity element, a subspace I such that I+W = W+I = W for all
Check that the set B = {0, 1} is a field under the operations listed above,
Give suitable operations to make the set {0, 1, 2} a field.
How many fundamental regions are there in one face of a speck of salt? (With a ruler, we can estimate that face is a square that is 0:1 cm on a side.)
In the graphite picture, imagine that we are interested in a point 5:67 Ångstroms over and 3:14 Ångstroms up from the origin.(a) Express that point in terms of the basis given for graphite.(b) How many hexagonal shapes away is this point from the origin?(c) Express that point in terms of a second
Give the locations of the atoms in the diamond cube both in terms of the basis, and in Ångstroms.
This illustrates how we could compute the dimensions of a unit cell from the shape in which a substance crystallizes. (a) Recall that there are 6.022 × 1023 atoms in a mole (this is Avogadro's number). From that, and the fact that platinum has a mass of 195.08 grams per mole, calculate the mass of
Here is a reasonable way in which a voter could have a cyclic preference. Suppose that this voter ranks each candidate on each of three criteria. (a) Draw up a table with the rows labeled 'Democrat', 'Republican', and 'Third', and the columns labeled 'character', 'experience', and 'policies'.
Compute the values in the table of decompositions.
Do the cancellations of opposite preference orders for the Political Science class's mock election. Are all the remaining preferences from the left three rows of the table or from the right?
The necessary condition that is proved above—a voting paradox can happen only if all three preference lists remaining after cancellation have the same spin—is not also sufficient.(a) Continuing the positive cycle case considered in the proof, use the two inequalities 0 < a - b + c and 0 <
A one-voter election cannot have a majority cycle because of the requirement that we've imposed that the voter's list must be rational. (a) Show that a two-voter election may have a majority cycle. (We consider the group preference a majority cycle if all three group totals are nonnegative or if
Let U be a subspace of R3. Prove that the set for all of vectors that are perpendicular to each vector in U is also subspace of R3. Does this hold if U is not a subspace?
Consider a projectile, launched with initial velocity v0, at an angle Î¸. To study its motion we may guess that these are the relevant quantities.(a) Show that {gt/v0, gx/v20, gy/v20, Î¸} is a complete set of dimensionless products. (One way to go is to find the appropriate
Conjectured that the infrared characteristic frequencies of a solid might be determined by the same forces between atoms as determine the solid's ordinary elastic behavior. The relevant quantities are these.Show that there is one dimensionless product. Conclude that, in any complete relationship
The torque produced by an engine has dimensional formula L2M1T-2. We may first guess that it depends on the engine's rotation rate (with dimensional formula L0M0T-1), and the volume of air displaced (with dimensional formula L3M0T0). (a) Try to find a complete set of dimensionless products. What
Dominoes falling make a wave. We may conjecture that the wave speed v depends on the spacing d between the dominoes, the height h of each domino, and the acceleration due to gravity g.(a) Find the dimensional formula for each of the four quantities.(b) Show that {∏1 = h/d, ∏2 = dg/v2 g is a
The advice about apples and ranges is not right. Consider the familiar equations for a circle C = 2πr and A = πr2. (a) Check that C and A have different dimensional formulas. (b) Produce an equation that is not dimensionally homogeneous (i.e., it adds apples and oranges) but is nonetheless true
Consider the isomorphism RepB(∙): P1 → R2 where B = (1, 1 + x). Find the image of each of these elements of the domain. (a) 3 - 2x, (b) 2 + 2x, (c) x
Verify, using Example 1.4 as a model, that the two correspondences given before the definition are isomorphism's. (a) Example 1.1 (b) Example 1.2
For the map f: P1 → R2 given byFind the image of each of these elements of the domain.(a) 3 – 2x(b) 2 + 2x(c) xShow that this map is an isomorphism.
Show that the natural map f1 from Example 1.5 is an isomorphism.
Show that the map t: P2 → P2 given by t(ax2 + bx + c) = bx2 - (a + c)x + a is an isomorphism.
Verify that this map is an isomorphism: h: R4 †’ M2Ã—2 given by
Decide whether each map is an isomorphism (if it is an isomorphism then prove it and if it isn't then state a condition that it fails to satisfy).(a) f: M2Ã—2 †’ R given by(b) f: M2Ã—2 †’ R4 given by (c) f: M2Ã—2 †’ P3 given by (d)
Show that the map f: R1 → R1 given by f(x) = x3 is one-to-one and onto. Is it an isomorphism?
Refer to Example 1.1. Produce two more isomorphism's (of course, you must also verify that they satisfy the conditions in the definition of isomorphism).
Refer to Example 1.2. Produce two more isomorphism's (and verify that they satisfy the conditions).
Show that, although R2 is not itself a subspace of R3, it is isomorphic to the xy-plane subspace of R3.
Find two isomorphism's between R16 and M4×4.
For what k is Mm×n isomorphic to Rk?
For what k is Pk isomorphic to Rn?
Prove that the map in Example 1.9, from P5 to P5 given by p(x) → p(x - 1), is a vector space isomorphism.
(Requires the subsection on Combining Subspaces, which is optional.) Suppose that V = V1 ⊕ V2 and that V is isomorphic to the space U under the map f. Show that U = f(V1) ⊕ f(U2).
True or false: between any n-dimensional space and Rn there is exactly one isomorphism.
Can a vector space be isomorphic to one of its (proper) subspaces?
Show that any isomorphism f: P0 → R1 has the form a → ka for some nonzero real number k.
These prove that isomorphism is an equivalence relation. (a) Show that the identity map id: V → V is an isomorphism. Thus, any vector space is isomorphic to itself. (b) Show that if f: V → W is an isomorphism then so is its inverse f-1: W → V. Thus, if V is isomorphic to W then also W is
Suppose that f: V → W preserves structure. Show that f is one-to-one if and only if the unique member of V mapped by f to W is V.
Suppose that f. V → W is an isomorphism. Prove that the set {1, . . . , k} ⊆ V is linearly dependent if and only if the set of images {f(1), . . . , f(k)} ⊆ W is linearly dependent.
Show that each type of map from Example 1.8 is an automorphism. (a) Dilation ds by a nonzero scalar s. (b) Rotation tθ through an angle θ. (c) Reflection fℓ over a line through the origin.
Produce an automorphism of P2 other than the identity map, and other than a shift map p(x) → p(x - k).
(a) Show that a function f: R1 †’ R1 is an automorphism if and only if it has the form x †’ kx for some k ‰ 0.(b) Let f be an automorphism of R1 such that f(3) = 7. Find f(- 2).(c) Show that a function f: R2 †’ R0 is an automorphism if and only if it has the
Refer to Lemma 1.10 and Lemma 1.11. Find two more things preserved by isomorphism.
We show that isomorphism's can be tailored to fit in that, sometimes, given vectors in the domain and in the range we can produce an isomorphism associating those vectors.(a) Let B = (1, 2, 3,) be a basis for P2 so that any ˆˆ P2 has a unique representation as = c11 + c22 + c33, which
This subsection shows that for any isomorphism, the inverse map is also an isomorphism. This subsection also shows that for a fixed basis B of an n-dimensional vector space V, the map RepB : V → Rn is an isomorphism. Find the inverse of this map.
(Requires the subsection on Combining Subspaces, which is optional.) Let U and W be vector spaces. Define a new vector space, consisting of the set U Ã— W = {(, ) | ˆˆ U and ˆˆ W} along with these operations.(1, 1) + (2, 2) = (1 + 2, 1 + 2) and r ˆ™ (, )
Show that the function from Theorem 2.3 is well-defined. Theorem 2.3 Vector spaces are isomorphic if and only if they have the same dimension.
For each, decide if it is a set of isomorphism class representatives. (a) {Ck | k ∈ N} (b) {Pk | k ∈ {- 1, 0, 1, . . .}} (c) {Mm×n | m, n ∈ N}
Let f be a correspondence between vector spaces V and W (that is, a map that is one-to-one and onto). Show that the spaces V and W are isomorphic via f if and only if there are bases B ⊂ V and D ⊂ W such that corresponding vectors have the same coordinates: RepB() = RepD(f()).
Consider the isomorphism RepB: P3 → R4. (a) Vectors in a real space are orthogonal if and only if their dot product is zero. Give a definition of orthogonality for polynomials. (b) The derivative of a member of P3 is in P3. Give a definition of the derivative of a vector in R4.
Does every correspondence between bases, when extended to the spaces, give an isomorphism? That is, suppose that V is a vector space with basis B = (1, . . . , n) and that f: B → W is a correspondence such that D = (f(1), . . . , f(n)) is basis for W. Must : V → W sending = c11 + . . .
Decide if the spaces are isomorphic. (a) R2, R4 (b) P5, R5 (c) M2×3, R6 (d) P5, M2×3 (e) M2×k, Ck
Decide if each h: R3 †’ R2 is linear.(a)(b) (c) (d)
Decide if each map h: M2Ã—2 †’ R is linear.(a)(b) (c) (d)
Show that these are homomorphisms. Are they inverse to each other? (a) d/dx: P3 → P2 given by a0 + a1x + a2x2 + a3x3 maps to a1 + 2a2x + 3a3x2 (b) ∫ P2 → P3 given by b0 + b1x + b2x2 maps to b0x + (b1/2)x2 + (b2/3)x3
Is (perpendicular) projection from R3 to the xz-plane a homomorphism? Projection to the yz-plane? To the x-axis? The y-axis? The z-axis? Projection to the origin?
Verify that each map is a homomorphism.(a) h: P3 †’ R2 given by(b) f: R2 †’ R3 given by
Show that, while the maps from Example 1.3 preserve linear operations, they are not isomorphism's.
For each linear map in the prior exercise, find the null space and nullity.
Stating that a function is 'linear' is different than stating that its graph is a line.(a) The function f1: R †’ R given by f1(x) = 2x - 1 has a graph that is a line. Show that it is not a linear function.(b) The function f2: R2 †’ R given by
What is the null space of the differentiation transformation d/dx: Pn → Pn? What is the null space of the second derivative, as a transformation of Pn? The k-th derivative?
Assume that h is a linear transformation of V and that (1, . . . , n) is a basis of V. Prove each statement. (a) If h(i) = for each basis vector then h is the zero map. (b) If h(i) = i for each basis vector then h is the identity map. (c) If there is a scalar r such that h(i) = r ∙ i for
Consider the vector space R+ where vector addition and scalar multiplication are not the ones inherited from R but rather are these: a + b is the product of a and b, and r a is the r-th power of a. (This was shown to be a vector space in an earlier exercise.) Verify that the natural logarithm map
Consider this transformation of R2.Find the image under this map of this ellipse.
Imagine a rope wound around the earth's equator so that it fits snugly (suppose that the earth is a sphere). How much extra rope must we add to raise the circle to a constant six feet off the ground?
Verify that this map h: R3 †’ Ris linear. Generalize.
Show that every homomorphism from R1 to R1 acts via multiplication by a scalar. Conclude that every nontrivial linear transformation of R1 is an isomorphism. Is that true for transformations of R2? Rn?
(a) Show that for any scalars a1, 1, . . . , am, n this map h. Rn †’ Rm is a homomorphism.(b) Show that for each i, the i-th derivative operator di/dxi is a linear transformation of Pn. Conclude that for any scalars ck, . . . , c0 this map is a linear transformation of that space.
Lemma 1.17 shows that a sum of linear functions is linear and that a scalar multiple of a linear function is linear. Show also that a composition of linear functions is linear.
Where f: V → W is linear, suppose that f(1) = 1, . . . , f(n) = n for some vectors 1, . . . , n from W. (a) If the set of 's is independent, must the set of 's also be independent? (b) If the set of 's is independent, must the set of 's also be independent? (c) If the set of 's spans
Generalize Example 1.16 by proving that for every appropriate domain and codomain the matrix transpose map is linear. What are the appropriate domains and codomains?
(a) Where , ∈ Rn, by definition the line segment connecting them is the set ℓ = {t ∙ + (1 - t) ∙ | t ∈ [0..1]}. Show that the image, under a homomorphism h, of the segment between and is the segment between h() and h(). (b) A subset of Rn is convex if, for any two points in
Let h: Rn → Rm be a homomorphism. (a) Show that the image under h of a line in Rn is a (possibly degenerate) line in Rm. (b) What happens to a k-dimensional linear surface?
Prove that the restriction of a homomorphism to a subspace of its domain is another homomorphism.
Assume that h: V → W is linear. (a) Show that the range space of this map {h() | ∈ V} is a subspace of the codomain W. (b) Show that the null space of this map { ∈ V | h() = W} is a subspace of the domain V. (c) Show that if U is a subspace of the domain V then its image {h() | ∈
Prove that for any transformation t: V → V that is rank one, the map given by composing the operator with itself t ◦ t: V → V satisfies t ◦ t = r ∙ t for some real number r.
Does Theorem 1.9 need that (1, . . . , n) is a basis? That is, can we still get a well-defined and unique homomorphism if we drop either the condition that the set of 's be linearly independent, or the condition that it span the domain?
Let V be a vector space and assume that the maps f1, f2: V †’ R1 are linear.(a) Define a map F: V †’ R2 whose component functions are the given linear ones.Show that F is linear. (b) Does the converse hold-is any linear map from V to R2 made up of two linear component maps to
Show that any linear map is the sum of maps of rank one.
Is 'is homomorphic to' an equivalence relation?
Show that the range spaces and null spaces of powers of linear maps t: V → V form descending V ⊇ R(t) ⊇ R(t2) ⊇ . . . and ascending {} ⊆ N (t) ⊆ N (t2) ⊆ . . . chains. Also show that if k is such that R(tk) = R(tk+1) then all following range spaces are equal. R(tk) = R(tk+1) =
Find the range space and the rank of each homomorphism.(a) h: P3 †’ R2 given by(b) f: R2 †’ R3 given by
Find the range space and rank of each map.(a) h: R2 †’ P3 given by(b) h: M2Ã—2 †’ R given by (c) h: M2Ã—2 †’ P2 given by (d) The zero map Z: R3 †’ R4.
Find the nullity of each map below. (a) h: R5 → R8 of rank five (b) h: P3 → P3 of rank one (c) h: R6 → R3, an onto map (d) h: M3×3 → M3×3, onto
For the homomorphism h: P3 → P3 given by h(a0 + a1x + a2x2 + a3x3) = a0 + (a0 + a1)x + (a2 + a3)x3 find these. (a) N (h) (b) h-1(2 - x3) (c) h-1 (1 + x2)
For the map f: R2 †’ R given bysketch these inverse image sets: f-1(- 3), f-1(0), and f-1(1).
Each of these transformations of P3 is one-to-one. For each, find the inverse. (a) a0 + a1x + a2x2 + a3x3 → a0 + a1x + 2a2x2 + 3a3x3 (b) a0 + a1x + a2x2 + a3x3 → a0 + a2x + a1x2 + a3x3 (c) a0 + a1x + a2x2 + a3x3 → a1 + a2x + a3x2 + a0x3 (d) a0 + a1x + a2x2 + a3x3 → a0 + (a0 + a1)x + (a0 +
Describe the null space and range space of a transformation given by → 2.
List all pairs (rank(h), nullity(h)) that are possible for linear maps from R5 to R3.
Find the nullity of this map h: Pn †’ R.
(a) Prove that a homomorphism is onto if and only if its rank equals the dimension of its codomain. (b) Conclude that a homomorphism between vector spaces with the same dimension is one-to-one if and only if it is onto.
Show that a linear map is one-to-one if and only if it preserves linear independence.
Corollary 2.17 says that for there to be an onto homomorphism from a vector space V to a vector space W, it is necessary that the dimension of W be less than or equal to the dimension of V. Prove that this condition is also sufficient; use Theorem 1.9 to show that if the dimension of W is less than
Recall that the null space is a subset of the domain and the range space is a subset of the codomain. Are they necessarily distinct? Is there a homomorphism that has a nontrivial intersection of its null space and its range space?

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