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linear algebra

Linear Algebra 1st Edition Jim Hefferon - Solutions

(a) Give a set that is closed under scalar multiplication but not addition.(b) Give a set closed under addition but not scalar multiplication.(c) Give a set closed under neither.
Subspaces are subsets and so we naturally consider how 'is a subspace of' interacts with the usual set operations.(a) If A, B are subspaces of a vector space, must their intersection A ∩ B be a subspace? Always? Sometimes? Never?(b) Must the union A ∪ B be a subspace?(c) If A is a subspace,
Does the span of a set depend on the enclosing space? That is, if W is a subspace of V and S is a subset of W (and so also a subset of V), might the span of S in W differ from the span of S in V?
Which of these subsets of the vector space of 2 Ã— 2 matrices are subspaces under the inherited operations? For each one that is a subspace, parametrize its description. For each that is not, give a condition that fails.(a)(b)(c)(d)
Decide whether each subset of R3 is linearly dependent or linearly independent.(a)(b)(c)(d)
Which of these subsets of P3 are linearly dependent and which are independent? (a) {3 - x + 9x2, 5 - 6x + 3x2, 1 + 1x - 5x2} (b) {-x2, 1 + 4x2} (c) {2 + x + 7x2, 3 - x + 2x2, 4 - 3x2} (d) {8 + 3x + 3x2, x + 2x2, 2 + 2x + 2x2, 8 - 2x + 5x2}
Determine if each set is linearly independent in the natural space.(a)(b) {(1 3 1), (-1 4 3), (-1 11 7)}(c)
Prove that each set {f, g} is linearly independent in the vector space of all functions from R+ to R. (a) f(x) = x and g(x) = 1/x (b) f(x) = cos(x) and g(x) = sin(x) (c) f(x) = ex and g(x) = ln(x)
Which of these subsets of the space of real-valued functions of one real variable is linearly dependent and which is linearly independent? (We have abbreviated some constant functions, e.g., in the first item, the '2' stands for the constant function f(x) = 2.) (a) {2, 4 sin2(x), cos2(x)} (b) {1,
Is the xy-plane subset of the vector space R3 linearly independent?
(a) Show that if the set is linearly independent then so is the set (b) What is the relationship between the linear independence or dependence of and the independence or dependence of
Example 1.10 shows that the empty set is linearly independent.(a) When is a one-element set linearly independent?(b) How about a set with two elements?
Show that if is linearly independent then so are all of its proper subsets: , and { }. Is that ‘only if’ also?
(a) Show that thisis a linearly independent subset of R3.(b) Show that 0is in the span of S by finding c1 and c2 giving a linear relationship.Show that the pair c1, c2 is unique.(c) Assume that S is a subset of a vector space and that is in [S], so that is a linear combination of vectors from S.
Prove that a polynomial gives rise to the zero function if and only if it is the zero polynomial. (Comment. This question is not a Linear Algebra matter but we often use the result. A polynomial gives rise to a function in the natural way: x → cnxn +∙ ∙ ∙+ c1x + c0.)
(a) Show that any set of four vectors in R2 is linearly dependent. (b) Is this true for any set of five? Any set of three? (c) What is the most number of elements that a linearly independent subset of R2 can have?
Is there a set of four vectors in R3 such that any three form a linearly independent set?
In R4 what is the biggest linearly independent set you can find? The smallest? The biggest linearly dependent set? The smallest? ('Biggest' and 'smallest' mean that there are no supersets or subsets with the same property.)
Linear independence and linear dependence are properties of sets. We can thus naturally ask how the properties of linear independence and dependence act with respect to the familiar elementary set relations and operations. In this body of this subsection we have covered the subset and superset
What is the interaction between the property of linear independence and the operation of union?(a) We might conjecture that the union S∪T of linearly independent sets is linearly independent if and only if their spans have a trivial intersection What is wrong with this argument for the
For Corollary 1.16,(a) Fill in the induction for the proof,(b) Give an alternate proof that starts with the empty set and builds a sequence of linearly independent subsets of the given finite set until one appears with the same span as the given set.
With a some calculation we can get formulas to determine whether or not a set of vectors is linearly independent.(a) Show that this subset of R2is linearly independent if and only if ad - bc 6= 0.(b) Show that this subset of R3is linearly independent iff aei + bfg + cdh - hfa - idb - gec ‰
(a) Prove that a set of two perpendicular nonzero vectors from Rn is linearly independent when n > 1. (b) What if n = 1? n = 0? (c) Generalize to more than two vectors.
Consider the set of functions from the interval (-1 . . . 1) ⊂ R to R. (a) Show that this set is a vector space under the usual operations. (b) Recall the formula for the sum of an infinite geometric series: 1 + x + x2 +. . . . . . = 1/(1 - x) for all x ∈ (-1. . . 1). Why does this not express
Show that, where S is a subspace of V, if a subset T of S is linearly independent in S then T is also linearly independent in V. Is that 'only if'?
Transpose each.(a)(b)(c)(d)(e) (-1 -2)
Find a basis for, and the dimension of, the solution set of this system. x1 - 4x2 + 3x3 - x4 = 0 2x1 - 8x2 + 6x3 - 2x4 = 0
Decide if each is a basis for P2. (a) (x2 - x + 1, 2x + 1, 2x - 1) (b) (x + x2, x - x2)
Decide if each is a basis for R3.(a)(b)(c)(d)
Represent the vector with respect to the basis.(a)(b) x2 + x3, D = h1, 1 + x, 1 + x + x2, 1 + x + x2 + x3) Š‚ P3(c)
Represent the vector with respect to each of the two bases.
Find a basis for P2, the space of all quadratic polynomials. Must any such basis contain a polynomial of each degree: degree zero, degree one, and degree two?
Find a basis for the solution set of this system. x1 - 4x2 + 3x3 - x4 = 0 2x1 - 8x2 + 6x3 - 2x4 = 0
Find a basis for M2×2, the space of 2 × 2 matrices.
Find a basis for each.(a) The subspace {a2x2 + a1x + a0 | a2 - 2a1 = a0 g of P2(b) The space of three-wide row vectors whose first and second components add to zero(c) This subspace of the 2 × 2 matrices f
Find a basis for each space, and verify that it is a basis.(a) The subspace M = {a + bx + cx2 + dx3 | a - 2b + c - d = 0} of P3.(b) This subspace of M2Ã—2.
Check Example 1.6. Example 1.6 Consider the space {a ∙ cos θ + b ∙ sin θ | a, b ∈ R} of functions of the real variable θ. This is a natural basis (cos θ, sin θ) = (1 ∙ cos θ + 0 ∙ sin θ, 0 ∙ cos θ + 1 ∙ sin θ). A more generic basis for this space is (cos θ - sin θ, 2 cos θ
Find the span of each set and then find a basis for that span. (a) {1 + x, 1 + 2x} in P2 (b) {2 - 2x, 3 + 4x2} in P2
Find a basis for each of these subspaces of the space P3 of cubic polynomials. (a) The subspace of cubic polynomials p(x) such that p(7) = 0 (b) The subspace of polynomials p(x) such that p(7) = 0 and p(5) = 0 (c) The subspace of polynomials p(x) such that p(7) = 0, p(5) = 0, and p(3) = 0 (d) The
What is the dimension of the vector space of functions f: S → R, under the natural operations, where the domain S is the empty set?
Find a basis for, and the dimension of, each space.(a)(b) the set of 5 Ã— 5 matrices whose only nonzero entries are on the diagonal (e.g., in entry 1; 1 and 2, 2, etc.)(c) {a0 + a1x + a2x2 + a3x3 | a0 + a1 = 0 and a2 - 2a3 = 0} Š‚ P3
Let be a basis for a vector space.(a) Show that is a basis when c1, c2, c3 ≠ 0. What happens when at least one ci is 0?(b) Prove that is a basis where
Find one vector that will make each into a basis for the space.(a)(b)(c) (x, 1 + x2, ) in P2
Where is a basis, show that in this equation each of the ci’s is zero. Generalize.
Prove that if U and W are both three-dimensional subspaces of R5 then U ∩ W is non-trivial. Generalize.
Theorem 1.12 shows that, with respect to a basis, every linear combination is unique. If a subset is not a basis, can linear combinations be not unique? If so, must they be?
A square matrix is symmetric if for all indices i and j, entry i, j equals entry j, i. (a) Find a basis for the vector space of symmetric 2 × 2 matrices. (b) Find a basis for the space of symmetric 3 ×3 matrices. (c) Find a basis for the space of symmetric n × n matrices.
We can show that every basis for R3 contains the same number of vectors. (a) Show that no linearly independent subset of R3 contains more than three vectors. (b) Show that no spanning subset of R3 contains fewer than three vectors. Recall how to calculate the span of a set and show that this method
One of the exercises in the Subspaces subsection shows that the setis a vector space under these operations.Find a basis.
Find a basis for, and the dimension of, M2×2, the vector space of 2 × 2 matrices.
Find the dimension of the vector space of matricessubject to each condition.(a) a, b, c, d ∈ R(b) a - b + 2c = 0 and d ∈ R(c) a + b + c = 0, a + b - c = 0, and d ∈ R
Find the dimension of this subspace of R2.
Find the dimension of each. (a) The space of cubic polynomials p(x) such that p(7) = 0 (b) The space of cubic polynomials p(x) such that p(7) = 0 and p(5) = 0 (c) The space of cubic polynomials p(x) such that p(7) = 0, p(5) = 0, and p(3) = 0 (d) The space of cubic polynomials p(x) such that p(7) =
What is the dimension of the span of the set {cos2 θ, sin2 θ, cos 2θ, sin 2θ}? This span is a subspace of the space of all real-valued functions of one real variable.
Find the dimension of C47, the vector space of 47-tuples of complex numbers.
What is the dimension of the vector space M3×5 of 3 × 5 matrices?
Show that this is a basis for R4.(We can use the results of this subsection to simplify this job.)
Refer to Example 2.12. (a) Sketch a similar subspace diagram for P2. (b) Sketch one for M2×2.
Prove that this is an infinite-dimensional space: the set of all functions f: R → R under the natural operations.
Show that is a basis if and only if there is no plane through the origin containing all three vectors.
Prove that any subspace of a finite dimensional space is finite dimensional.
A basis for a space consists of elements of that space. So we are naturally led to how the property 'is a basis' interacts with operations ⊂ and ∩ and ∪. (Of course, a basis is actually a sequence in that it is ordered, but there is a natural extension of these operations.) (a) Consider first
Consider how 'dimension' interacts with 'subset'. Assume U and W are both subspaces of some vector space, and that U ⊂ W. (a) Prove that dim(U) 6 dim(W). (b) Prove that equality of dimension holds if and only if U = W. (c) Show that the prior item does not hold if they are infinite-dimensional.
For any vector in Rn and any permutation σ of the numbers 1, 2, . . . , n (that is, σ is a rearrangement of those numbers into a new order), define σ() to be the vector whose components are vσ(1), vσ(2), . . . , and vσ(n) (where σ(1) is the first number in the rearrangement, etc.). Now
Decide if the vector is in the row space of the matrix.(a)(b)
Decide if the vector is in the column space.(a)(b)
Decide if the vector is in the column space of the matrix.(a)(b)(c)
Find a basis for the row space of this matrix.
Find the rank of each matrix.(a)(b)(c)(d)
Give a basis for the column space of this matrix. Give the matrix's rank.
Find a basis for the span of each set.(a) {(1 3), (-1 3), (1 4), (2 1) g Š‚ M1Ã—2(b)(c) {1 + x, 1 - x2, 3 + 2x - x2 g} P3(d)
Give a basis for the span of each set, in the natural vector space.(a)(b) {x + x2, 2 - 2x, 7, 4 + 3x + 2x2}
Which matrices have rank zero? Rank one?
Given a, b, c ˆˆ R, what choice of d will cause this matrix to have the rank of one?
Find the column rank of this matrix.
Show that a linear system with at least one solution has at most one solution if and only if the matrix of coefficients has rank equal to the number of its columns.
Give an example to show that, despite that they have the same dimension, the row space and column space of a matrix need not be equal. Are they ever equal?
Show that the set {(1,-1, 2,-3), (1, 1, 2, 0), (3,-1, 6,-6)} does not have the same span as {(1, 0, 1, 0), (0, 2, 0, 3)}. What, by the way, is the vector space?
Show that this set of column vectorsis a subspace of R3. Find a basis.
Show that the transpose operation is linear: (rA + sB)T = rAT + sBT for r, s ∈ R and A, B ∈ Mm×n.
In this subsection we have shown that Gaussian reduction finds a basis for the row space. (a) Show that this basis is not unique-different reductions may yield different bases. (b) Produce matrices with equal row spaces but unequal numbers of rows. (c) Prove that two matrices have equal row spaces
Show that the row rank of an mn matrix is at most m. Is there a better bound?
Show that the rank of a matrix equals the rank of its transpose.
True or false: the column space of a matrix equals the row space of its transpose.
We have seen that a row operation may change the column space. Must it?
Prove that a linear system has a solution if and only if that system's matrix of coefficients has the same rank as its augmented matrix.
An m × n matrix has full row rank if its row rank is m, and it has full column rank if its column rank is n. (a) Show that a matrix can have both full row rank and full column rank only if it is square. (b) Prove that the linear system with matrix of coefficients A has a solution for any d1, . . .
What is the relationship between rank(A) and rank(-A)? Between rank(A) and rank(kA)? What, if any, is the relationship between rank(A), rank(B), and rank(A + B)?
Decide if R2 is the direct sum of each W1 and W2.(a)(b)(c) W1 = R2, W2 = {}(d)(e)
Show that R3 is the direct sum of the xy-plane with each of these.(a) The z-axis(b) The line
Is P2 the direct sum of {a + bx2 | a, b ∈ R} and {cx | c ∈ R}?
In Pn, the even polynomials are the members of this setand the odd polynomials are the members of this set.Show that these are complementary subspaces.
Which of these subspaces of R3W1: the x-axis, W2: the y-axis, W3: the z-axis,W4: the plane x + y + z = 0, W5: the yz-planecan be combined to(a) Sum to R3?(b) Direct sum to R3?
Can R4 be decomposed as a direct sum in two different ways? Can R1?
This exercise makes the notation of writing '+' between sets more natural. Prove that, where W1, . . . . . ,Wk are subspaces of a vector space,and so the sum of subspaces is the subspace of all sums.
Prove that if V = W1 ⊕ . . . . . . . ⊕ Wk then Wi ∩ Wj is trivial whenever i ≠ j. This shows that the first half of the proof of Lemma 4.15 extends to the case of more than two subspaces. (Example 4.19 shows that this implication does not reverse, the other half does not extend.)
Recall that no linearly independent set contains the zero vector. Can an independent set of subspaces contain the trivial subspace?
Does every subspace have a complement?
Let W1,W2 be subspaces of a vector space.(a) Assume that the set S1 spans W1, and that the set S2 spans W2. Can S1 ∪ S2 span W1 +W2? Must it?(b) Assume that S1 is a linearly independent subset of W1 and that S2 is a linearly independent subset of W2. Can S1 ⊂ S2 be a linearly independent subset
When we decompose a vector space as a direct sum, the dimensions of the subspaces add to the dimension of the space. The situation with a space that is given as the sum of its subspaces is not as simple. This exercise considers the two-subspace special case.(a) For these subspaces of M2×2 find W1

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