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linear algebra

Linear Algebra 1st Edition Jim Hefferon - Solutions

Prove that the image of a span equals the span of the images. That is, where h: V → W is linear, prove that if S is a subset of V then h([S]) equals [h(S)]. This generalizes Lemma 2.1 since it shows that if U is any subspace of V then its image {h() | ∈ U} is a subspace of W, because the
(a) Prove that for any linear map h: V †’ W and any ˆˆ W, the set h-1() has the form{ + | ˆˆ N (h)}for ˆˆ V with h() = (if h is not onto then this set may be empty). Such a set is a coset of N (h) and we denote it as + N (h).(b) Consider the map t: R2
Let h. V → R be a homomorphism, but not the zero homomorphism. Prove that if (1, . . . , n) is a basis for the null space and if ∈ V is not in the null space then (,1, . . . , n) is a basis for the entire domain V.
Show that for any space V of dimension n, the dual space L(V, R) = {h: V → R | h is linear} is isomorphic to Rn. It is often denoted V*. Conclude that V* ≅ V.
Let h: P3 → P4 be given by p(x) → x ∙ p(x). Which of these are in the null space? Which are in the range space? (a) x3 (b) 0 (c) 7 (d) 12x - 0.5x3 (e) 1 + 3x2 - x3
Multiply the matrixby each vector, or state "not defined." (a) (b) (c)
Perform, if possible, each matrix-vector multiplication.(a)(b) (c)
Solve this matrix equation.
For a homomorphism from P2 to P3 that sends 1 → 1 + x, x → 1 + 2x and x2 → x - x3 where does 1 - 3x + 2x2 go?
Assume that h: R2 †’ R3 is determined by this action.Using the standard bases, find (a) The matrix representing this map; (b) A general formula for h().
Represent the homomorphism h: R3 †’ R2 given by this formula and with respect to these bases.
Let d/dx: P3 → P3 be the derivative transformation. (a) Represent d/dx with respect to B, B where B = (1, x, x2, x3). (b) Represent d/dx with respect to B, D where D = (1, 2x, 3x2, 4x3).
Represent each linear map with respect to each pair of bases. (a) d/dx: Pn → Pn with respect to B, B where B = (1, x, . . . , xn), given by a0 + a1x + a2x2 + . . . + anxn → a1 + 2a2x + . . . + nanxn-1 (b) ∫: Pn → Pn+1 with respect to Bn, Bn+1 where Bi = (1, x, . . . , xi), given by a0 + a1x
Represent the identity map on any nontrivial space with respect to B, B, where B is any basis.
Represent, with respect to the natural basis, the transpose transformation on the space M2×2 of 2 × 2 matrices.
Assume that B = (1, 2, 3, 4) is a basis for a vector space. Represent with respect to B, B the transformation that is determined by each. (a) 1 → 2, 2 → 3, 3, → 4, 4 → . (b) 1 → 2, 2 → , 3, → 4, 4 → . (c) 1 → 2, 2 → 3, 3, → , 4 → .
Example 1.9 shows how to represent the rotation transformation of the plane with respect to the standard basis. Express these other transformations also with respect to the standard basis. (a) The dilation map ds, which multiplies all vectors by the same scalar s (b) The reflection map fℓ, which
Consider a linear transformation of R2 determined by these two.(a) Represent this transformation with respect to the standard bases. (b) Where does the transformation send this vector? (c) Represent this transformation with respect to these bases. (d) Using B from the prior item, represent the
Suppose that h: V → W is one-to-one so that by Theorem 2.20, for any basis B = (1, . . . , n) ⊂ V the image h(B) = h((1), . . . , h(n)) is a basis for W. (a) Represent the map h with respect to B, h(B). (b) For a member of the domain, where the representation of has components c1, . . .
Give a formula for the product of a matrix and i, the column vector that is all zeroes except for a single one in the i-th position.
For each vector space of functions of one real variable, represent the derivative transformation with respect to B, B. (a) {a cos x + b sin x | a, b ∈ R}, B = (cos x, sin x) (b) {aex + be2x | a, b ∈ R}, B = (ex, e2x) (c) {a + bx + cex + dxex | a, b, c, d ∈ R}, B = (1, x, ex, xex)
Find the range of the linear transformation of R2 represented with respect to the standard bases by each matrix.(a)(b) (c) A matrix of the form
Can one matrix represent two different linear maps? That is, can RepB,D(h) = Rep, (ĥ)?
Prove Theorem 1.4.Assume that V and W are vector spaces of dimensions n and m with bases B and D, and that h: V †’ W is a linear map. If h is represented byand ˆˆ V is represented by then the representation of the image of is this.
Example 1.9 shows how to represent rotation of all vectors in the plane through an angle θ about the origin, with respect to the standard bases. (a) Rotation of all vectors in three-space through an angle θ about the x-axis is a transformation of R3. Represent it with respect to the standard
(Schur's Triangularization Lemma) (a) Let U be a subspace of V and fix bases BU ⊆ BV. What is the relationship between the representation of a vector from U with respect to BU and the representation of that vector (viewed as a member of V) with respect to BV? (b) What about maps? (c) Fix a basis
Let h be the linear map defined by this matrix on the domain P1 and codomain R2 with respect to the given bases.What is the image under h of the vector = 2x - 1?
Decide if each vector lies in the range of the map from R3 to R2 represented with respect to the standard bases by the matrix.(a)(b)
Consider this matrix, representing a transformation of R2, and these bases for that space.(a) To what vector in the codomain is the first member of B mapped? (b) The second member? (c) Where is a general vector from the domain (a vector with components x and y) mapped? That is, what transformation
What transformation of F = {a cos Î¸ + b sin Î¸ | a, b ˆˆ R} is represented with respect to B = (cos Î¸ - sin Î¸, sin Î¸) and D = (cos Î¸ + sin Î¸, cos Î¸) by this matrix?
Decide whether 1 + 2x is in the range of the map from R3 to P2 represented with respect to Îµ3 and (1, 1 + x2, x) by this matrix.
Example 2.11 gives a matrix that is nonsingular and is therefore associated with maps that are nonsingular. (a) Find the set of column vectors representing the members of the null space of any map represented by this matrix. (b) Find the nullity of any such map. (c) Find the set of column vectors
Take each matrix to represent h: Rm †’ Rn with respect to the standard bases. For each (i) state m and n. Then set up an augmented matrix with the given matrix on the left and a vector representing a range space element on the right(e.g., if the codomain is R3 then in the right-hand
Use the method from the prior exercise on each.(a)(b) Verify that the map represented by this matrix is an isomorphism.
This is an alternative proof of Lemma 2.9. Given an n × n matrix H, fix a domain V and codomain W of appropriate dimension n, and bases B, D for those spaces, and consider the map h represented by the matrix. (a) Show that h is onto if and only if there is at least one RepB() associated by H with
Let V be an n-dimensional space with bases B and D. Consider a map that sends, for ∈ V, the column vector representing with respect to B to the column vector representing with respect to D. Show that map is a linear transformation of Rn.
Example 2.3 shows that changing the pair of bases can change the map that a matrix represents, even though the domain and codomain remain the same. Could the map ever not change? Is there a matrix H, vector spaces V and W, and associated pairs of bases B1,D1 and B2,D2 (with B1 ≠ B2 or D1 ≠ D2
Describe geometrically the action on R2 of the map represented with respect to the standard bases Îµ2, Îµ2 by this matrix.Do the same for these.
The fact that for any linear map the rank plus the nullity equals the dimension of the domain shows that a necessary condition for the existence of a homomorphism between two spaces, onto the second space, is that there be no gain in dimension.That is, where h: V †’ W is onto, the
Let V be an n-dimensional space and suppose that ∈ Rn. Fix a basis B for V and consider the map h : V → R given → ∙ RepB() by the dot product. (a) Show that this map is linear. (b) Show that for any linear map g: V → R there is an ∈ Rn such that g = h. (c) In the prior item we
Prove each, assuming that the operations are defined, where G, H, and J are matrices, where Z is the zero matrix, and where r and s are scalars. (a) Matrix addition is commutative G + H = H + G. (b) Matrix addition is associative G + (H + J) = (G + H) + J. (c) The zero matrix is an additive
Fix domain and codomain spaces. In general, one matrix can represent many different maps with respect to different bases. However, prove that a zero matrix represents only a zero map. Are there other such matrices?
Let V and W be vector spaces of dimensions n and m. Show that the space L(V,W) of linear maps from V to W is isomorphic to Mm×n.
Show that it follows from the prior questions that for any six transformations t1, . . . , t6 ∙ R2 → R2 there are scalars c1, . . . , c6 ∈ R such that c1t1 + . . . + c6t6 is the zero map.
The trace of a square matrix is the sum of the entries on the main diagonal (the 1, 1 entry plus the 2, 2 entry, etc., we will see the significance of the trace in Chapter Five). Show that trace(H + G) = trace(H) + trace(G). Is there a similar result for scalar multiplication?
Recall that the transpose of a matrix M is another matrix, whose i, j entry is the j, i entry of M. Verify these identities. (a) (G + H)T = GT + HT (b) (r ∙ H)T = r ∙ HT
A square matrix is symmetric if each i, j entry equals the j, i entry, that is, if the matrix equals its transpose. (a) Prove that for any square H, the matrix H + HT is symmetric. Does every symmetric matrix have this form? (b) Prove that the set of n × n symmetric matrices is a subspace of Mn×n.
(a) How does matrix rank interact with scalar multiplication-can a scalar product of a rank n matrix have rank less than n? Greater? (b) How does matrix rank interact with matrix addition-can a sum of rank n matrices have rank less than n? Greater?
Find the system of equations resulting from starting with h1,1x1 + h1,2x2 + h1,3x3 = d1 h2,1x1 + h2,2x2 + h2,3x3 = d2 and making this change of variable (i.e., substitution). x1 = g1,1y1 + g1,2y2 x2 = g2,1y1 + g2,2y2 x3 = g3,1y1 + g3,2y2
Consider the two linear functions h: R3 †’ P2 and g: P2 †’ M2Ã—2 given as here.Use these bases for the spaces. (a) Give the formula for the composition map g —¦ h: R3 †’ M2Ã—2 derived directly from the above definition. (b) Represent
Wherecompute or state 'not defined'. (a) AB (b) (AB)C (c) BC (d) A(BC)
Represent the derivative map on Pn with respect to B, B where B is the natural basis (1, x, . . . , xn). Show that the product of this matrix with itself is defined, what map does it represent?
Match each type of matrix with all these descriptions that could fit: (i) Can be multiplied by its transpose to make a 1 × 1 matrix, (ii) Is similar to the 3 × 3 matrix of all zeros, (iii) Can represent a linear map from R3 to R2 that is not onto, (iv) Can represent an isomorphism from R3 to
Show that composition of linear transformations on R1 is commutative. Is this true for any one-dimensional space?
Which products are defined? (a) 3 × 2 times 2 × 3 (b) 2 × 3 times 3 × 2 (c) 2 × 2 times 3 × 3 (d) 3 × 3 times 2 × 2
Give the size of the product or state "not defined". (a) a 2 × 3 matrix times a 3 × 1 matrix (b) a 1 × 12 matrix times a 12 × 1 matrix (c) a 2 × 3 matrix times a 2 × 1 matrix (d) a 2 × 2 matrix times a 2 × 2 matrix
(a) How does matrix multiplication interact with scalar multiplication: is r(GH) = (rG)H? Is G(rH) = r(GH)? (b) How does matrix multiplication interact with linear combinations: is F(rG + sH) = r(FG) + s(FH)? Is (rF + sG)H = rFH + sGH?
We can ask how the matrix product operation interacts with the transpose operation. (a) Show that (GH)T = HTGT. (b) A square matrix is symmetric if each i, j entry equals the j, i entry, that is, if the matrix equals its own transpose. Show that the matrices HHT and HTH are symmetric.
Rotation of vectors in R3 about an axis is a linear map. Show that linear maps do not commute by showing geometrically that rotations do not commute.
In the proof of Theorem 2.12 we used some maps. What are the domains and codomains? Theorem 2.12 If F, G, and H are matrices, and the matrix products are defined, then the product is associative (FG)H = F(GH) and distributes over matrix addition F(G + H) = FG + FH and (G + H)F = GF + HF.
How does matrix rank interact with matrix multiplication? (a) Can the product of rank n matrices have rank less than n? Greater? (b) Show that the rank of the product of two matrices is less than or equal to the minimum of the rank of each factor.
Is 'commutes with' an equivalence relation among n × n matrices?
(We will use this exercise in the Matrix Inverses exercises.) Here is another property of matrix multiplication that might be puzzling at first sight. (a) Prove that the composition of the projections πx, πy: R3 → R3 onto the x and y axes is the zero map despite that neither one is itself the
Show that, for square matrices, (S + T)(S - T) need not equal S2 - T2.
Represent the identity transformation id : V → V with respect to B, B for any basis B. This is the identity matrix I. Show that this matrix plays the role in matrix multiplication that the number 1 plays in real number multiplication: HI = IH = H (for all matrices H for which the product is
Compute, or state "not defined".(a)(b) (c) (d)
(a) Prove that for any 2 Ã— 2 matrix T there are scalars c0, . . . , c4 that are not all 0 such that the combination c4T4 + c3T3 + c2T2 + c1T + c0I is the zero matrix (where I is the 2 Ã— 2 identity matrix, with 1's in its 1, 1 and 2, 2 entries and zeroes elsewhere, see
The infinite-dimensional space P of all finite-degree polynomials gives a memorable example of the non-commutativity of linear maps. Let d/dx: P †’ P be the usual derivative and let s: P †’ P be the shift map.Show that the two maps don't commute d/dx —¦ s
Recall the notation for the sum of the sequence of numbers a1, a2, . . . , an.In this notation, the I, j entry of the product of G and H is this. Using this notation, (a) Reprove that matrix multiplication is associative; (b) Reprove Theorem 2.7.
Describe the product of two diagonal matrices.
Show that if G has a row of zeros then GH (if defined) has a row of zeros. Does that work for columns?
Show that the set of unit matrices forms a basis for Mn×m.
Find the formula for the n-th power of this matrix.
The trace of a square matrix is the sum of the entries on its diagonal (its significance appears in Chapter Five). Show that Tr(GH) = Tr(HG).
A square matrix is upper triangular if its only nonzero entries lie above, or on, the diagonal. Show that the product of two upper triangular matrices is upper triangular. Does this hold for lower triangular also?
A square matrix is a Markov matrix if each entry is between zero and one and the sum along each row is one. Prove that a product of Markov matrices is Markov.
Give an example of two matrices of the same rank and size with squares of differing rank.
On a computer multiplications have traditionally been more costly than additions, so people have tried to in reduce the number of multiplications used to compute a matrix product. (a) How many real number multiplications do we need in the formula we gave for the product of a m × r matrix and a r
If A and B are square matrices of the same size such that ABAB = 0, does it follow that BABA = 0?
Demonstrate these four assertions to get an alternate proof that column rank equals row rank. (a) ∙ = 0 iff = . (b) A = iff ATA = . (c) dim(R(A)) = dim(R(ATA)). (d) col rank(A) = col rank(AT) = row rank(A).
Prove (where A is an nn matrix and so defines a transformation of any n-dimensional space V with respect to B, B where B is a basis) that dim(R(A) ∩ N (A)) = dim(R(A)) - dim(R(A2)). Conclude (a) N (A) ⊂ R(A) iff dim(N (A)) = dim(R(A)) - dim(R(A2)), (b) R(A) ⊆ N (A) iff A2 = 0, (c) R(A) = N
Predict the result of each multiplication by an elementary reduction matrix, and then check by multiplying it out.(a)(b) (c) (d) (e)
Predict the result of each multiplication by a diagonal matrix, and then check by multiplying it out.(a)(b)
Produce each. (a) a 3 × 3 matrix that, acting from the left, swaps rows one and two (b) a 2 × 2 matrix that, acting from the right, swaps column one and two
Show how to use matrix multiplication to bring this matrix to echelon form.
Find the product of this matrix with its transpose.
The need to take linear combinations of rows and columns in tables of numbers arises often in practice. For instance, this is a map of part of Vermont and New York.In part because of Lake Champlain, there are no roads directly connecting some pairs of towns. For instance, there is no way to go from
Express this nonsingular matrix as a product of elementary reduction matrices.
Expressas the product of two elementary reduction matrices.
Prove that the diagonal matrices form a subspace of Mn×n. What is its dimension?
Does the identity matrix represent the identity map if the bases are unequal?
Show that every multiple of the identity commutes with every square matrix. Are there other matrices that commute with all square matrices?
Show that the product of a permutation matrix and its transpose is an identity matrix.
Show that if the first and second rows of G are equal then so are the first and second rows of GH. Generalize.
Supply the intermediate steps in Example 4.9.
Use Corollary 4.11 to decide if each matrix has an inverse.Corollary 4.11The inverse for a 2 Ã— 2 matrix exists and equalsif and only if ad - bc ‰ 0. (a) (b) (c)
For each invertible matrix in the prior problem, use Corollary 4.11 to find its inverse.Corollary 4.11The inverse for a 2 Ã— 2 matrix exists and equalsif and only if ad - bc ‰ 0.
Find the inverse, if it exists, by using the Gauss-Jordan Method. Check the answers for the 2 Ã— 2 matrices with Corollary 4.11.(a)(b) (c) (d) (e) (f)
What matrix has this one for its Î¸ inverse?
How does the inverse operation interact with scalar multiplication and addition of matrices? (a) What is the inverse of rH? (b) Is (H + G)-1 = H-1 + G-1?

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