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Elementary Linear Algebra with Applications 9th edition Howard Anton, Chris Rorres - Solutions

(a) What does the inequality || x || < 1 tell you about the location of the point x in the plane? (b) Write down an inequality that describes the set of points that lie outside the circle of radius 1, centered at the point x0.
Let u = (2, - 2, 3), v = (1, - 3, 4), w = (3, 6, - 4). In each part, evaluate the expression. (a) || u + v || (b) || - 2u || + 2|| u || (c) 1 / || w || w
Let v = (- 1, 2, 5). Find all scalars k such that || kv || = 4.
Fin u ∙ v. (a) u = (2, 3), v = (5, - 7) (b) u = (1, -5,)4, v = (3, 3, 3)
Find a unit vector that is orthogonal to both u = (1, 0, 1) and Y = (0, 1, 1).
Establish the identity u ∙ v = 1/4 || u + v || - 1/4 || u - v ||2.
Let i, j, and k be unit vectors along the positive x, y, and z axes of a rectangular coordinate system in 3-space. If v = (a, b, c) is a nonzero vector, then the angles Î±, Î², and Î³ between v and the vectors i, j, and k, respectively, are called the direction
Referring to Exercise 21, show that two nonzero vectors, v1 and v2, in 3-space are perpendicular if and only if their direction cosines satisfy cos α1 cos α2 + cos β1 cos β2 + cos γ1 cos γ2 = 0
Show that if v is orthogonal to both w1 and w2, then v is orthogonal to k1w1 + k2w2 for all scalars k1 and k2.
In each part, something is wrong with the expression. What? (a) u ∙ (v ∙ w) (b) (u ∙ v) + w (c) ||u ∙ v|| (d) k ∙ (u + w)
If u ≠ 0, is it valid to cancel u from both sides of the equation u ∙ v = u ∙ w and conclude that y = w? Explain your reasoning.
Determine whether u and v make an acute angle, make an obtuse angle, or are orthogonal. (a) u = (6, 1,4), v = (2, 0, - 3) (b) u = (0, 0, - 1), v = (1, 1, 1)
Suppose that u and v are orthogonal vectors in 2-space or 3-space. What famous theorem is described by the equation ||u + v||2 = ||u||2 + ||v||2? Draw a picture to support your answer.
In each part of Exercise 4, find the vector component of u orthogonal to a. (a) u = (6, 2), a = (3, -9) (b) u = (3, 1, - 7), a = (1, 0, 5)
Let u = (3, 2, - 1), v = (0, 2, - 3), and w = (2, 6, 7). Compute (a) v × w (b) (u × v) × w (c) u × (v - 2w)
Determine whether u, v, and w lie in the same plane when positioned so that their initial points coincide. u = (- l, - 2, l),v = (3, 0, - 2),w = (5, - 4,0)
Simplify (u + v) × (u - v).
Find the area of the triangle having vertices A(1, 0, 1), 5(0, 2, 3), and C(2, 1, 0).
Use the result of Exercise 18 to find the distance between the point P and the line through the points A and P(- 3, 1, 2), A(1, 1, 0), B( - 2, 3, - 4)
Consider the parallelepiped with sides u = (3, 2, 1), v = (1, 1, 2), and w = (1, 3, 3). Find the angle between u and the plane containing the face determined by v and w.
Let u, v, and w be nonzero vectors in 3-space with the same initial point, but such that no two of them are collinear. Show that (a) u × (v × w) lies in the plane determined by v and w (b) (u × v) × w lies in the plane determined by u and v
Prove: If a, b, c, and d he in the same plane, then (a × b) × (c × d) = 0.
Find the area of the parallelogram determined by u and (a) u = (1, - 1, 2), v = (0, 3, 1) (b) u = (3, - 1,4), v = (6, - 2, 8)
Use the result of Exercise 30 to find the volume of the tetrahedron with vertices P, Q, R, S. P(0, 0, 0), Q(1, 2, - 1), R(3, 4, 0), S( - 1, - 3, 4)
Prove parts (c) and (d) of Theorem 3.4.2. (c) (u + v) × w = (n × w) + (v × w) (d) k(u × v) = (ku) × v = u × (kv)
(a) Suppose that 11 and v are noncollinear vectors with their initial points at the origin in 3-spaci Make a sketch that illustrates how w = v × (u × v) is oriented in relation to u and v. (b) For w as in part (a), what can you say about the values of u ∙ w and y ∙ w? Explain your reasoning.
Something is wrong with one of the following expressions. Which one is it and what is wrong? u ∙ (v × w), u × v × w, (u ∙ v) × w
Find a vector v that is orthogonal to the vector u = (2, - 3, 5).
Suppose that u ∙ (v x w) = 3- Find (u × w) ∙ v
Find parametric equations for the line of intersection of the given planes. 7x - 2y + 3z = - 2 and - 3x + y + 2z + 5 = 0
Determine whether the planes are parallel. (a) (- 1, 2, 4) - (x - 5, y + 3, z - 7) = 0; (2, - 4, - 8) - (x + 3, y + 5, z - 9) = 0 (b) (3, 0, - 1) - (x + 1, y - 2, z - 3) = 0; (- 1, 0, 3) - (x + 1, y - z, z - 3) = 0
Find an equation for the plane through (- 2, 1, 7) that is perpendicular to the line x - 4 = 2t, y + 2 = 3t, z = - 5t.
Find an equation of the plane that contains the point (x0, y0, z0) and is Parallel to the xy-plane
Find an equation for the plane that passes through the point (3, - 6, 7) and is parallel to the plane 5x - 2y + z - 5 = 0
Show that the points (-1, - 2, - 3), (- 2, 0, 1), (- 4, - 1, - 1), and (2, 0, 1) lie in the same plane.
Find an equation for the plane through (- 2, 1, 5) that is perpendicular to the planes 4x - 2y + 2z = - 1 and 3x + 3y - 6z = 5.
Find an equation for the plane that contains the point (1, - 1, 2) and the line x = t, y = t + 1, z = - 3 + 2t.
Find an equation for the plane, each of whose points is equidistant from (- 1, - 4, - 2) and (0, - 2, 2).
Show that the lines x - 3 = 4t, y - 4 = t, z - 1 = 0 (- ∞ < t < + ∞) And x + 1 = 12t, y - 7 = 6t, z - 5 = 3t (- ∞ < t < + ∞)
Find parametric equations for the line of intersection of the planes - 3x + 2y + z = - 5 and 7x + 3y - 2z = - 2
Find the distance between the point and the plane. (- 1, 2, 1); 2x + 3y - 4z = 1
Find the distance between the line x = 2t - 1, y = 2 - t, z = t and each of the following points. (a) (0, 0, 0) (b) (2, 0, - 5) (c) (2, 1, 1)
Two intersecting planes in 3-space determine two angles of intersection: an acute angle (0 x = 0 and 2x - y + z - 4 = 0
What do the lines r = r0 + tv and r = r0 - tv have in common? Explain.
Let r1 and r2 be vectors from the origin to the points P1(x1, y1, z1) and P2(x2, y2, z2), respectively. What does the equation r = (1 - t)r1 + tr2 (0 ≤ t ≤ 1) represent geometrically? Explain your reasoning.
Determine whether the planes are parallel. (a) 4x - y + 2z = 5 and 7x - 3y + 4z = 8 (b) x - 4y - 3z - 2 = 0 and 3x - 12y - 9z - 1 = 0
Find the Euclidean distance between u and v. (a) u = (1, - 2), v = (2, 1) (b) u = (0, - 2, - 1, 1), v = (- 3, 2, 4, 4)
In each part, verify that the Cauchy-Schwarz inequality holds. (a) u = (3, 2), v = (4, - 1) (b) u = (0, - 2, 2, 1), v = (- 1, - 1, 1, 1)
Determine whether the two lines r = (3, 2, 3, -1) + t(4, 6, 4, -2) and r = (0, 3, 5, 4) + s(l, - 3, - 4, - 2) intersect in R4.
Prove: If u, v, and w are vectors in Rn and k is any scalar, then (a) u ∙ (kv) = k(u ∙ v) (b) u ∙ (v + w) = u ∙ v + u ∙ w
Let u1 = (- 1, 3, 2, 0), u2 = (2, 0, 4, - 1), u3 = (7, 1, 1, 4), and u4 = (6, 3, 1, 2). Find scalars c1, c2, c3, and c4 such that c1u1 + c2u2 + c3u3 + c4u4 = (0, 5, 6, - 3).
Indicate whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. (a) If ||u + v ||2 = ||u||2 + ||v||2 then u and v are orthogonal. (b) If u is orthogonal to v and w, then u is orthogonal to v + w (c) If u is orthogonal to v +
Find the Euclidean inner product u ∙ v. (a) u = (2, 5),v = (- 4, 3) (b) u = (3, 1,4, - 5), v = (2, 2, - 4, - 3)
Use matrix multiplication to find the image of the vector (- 2, 1, 2) if it is rotated 45° about the j-axis
Use matrix multiplication to find the image of the vector (- 2, 1, 2) if it is rotated - 45° about the y-axis
Find the standard matrix for the stated composition of linear operators on R2. (a) A rotation of 60°, followed by an orthogonal projection on the x-axis, followed by a reflection about the line y = x (b) A rotation of 15°, followed by a rotation of 105°, followed by a rotation of 60°.
Determine whether T1 ◦ T2 = T2 ◦ T1. T1: R3 → R3 is a dilation by a factor k, and T2: R3 → R3 is the rotation about the z axis through an angle θ.
By inspection, determine whether the linear operator is one-to-one. (a) The orthogonal projection on the x-axis in R2. (b) The reflection about the y-axis in R2.
Use Theorem 4.3.3 to find the standard matrix for T: R2 → R2 from the images of the standard basis vectors. (a) T: R2 → R2 projects a vector orthogonally onto the x-axis and then reflects that vector about the y-axis. (b) T: R2 → R2 reflects a vector about the line y = x and then reflects
Use the result in Example 6 to find the orthogonal projection of x onto the line through the origin that makes an angle 0 with the positive x-axis. x = (- 1, 2); θ = 45°
Follow the directions of Exercise 18. (a) T: R3 → R3 is the reflection about the yz-plane. (b) T: R3 → R3 is the orthogonal projection on the xz-plane. (c) T: R3 → R3 is the dilation by a factor of 2. (d) T: R3 → R3 is a rotation of π / 4 about the r-axis.
Show that T(x, y) = (0, 0) defines a linear operator on |R2| but T(x, y) = (1, 1) does not.
Let T: R2 †’ R2 be the linear operator that reflects each vector about / (see the accompanying figure).(a) Use the method of Example 6 to find the standard matrix for T.(b) Find the reflection of the vector x = (1, 5) about the line / through the origin that makes an angle of
Indicate whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. (a) If Rn maps Rm into and T(0) = 0, then T is linear. (b) If T: Rn → Rm is a one-to-one linear transformation, then there are no distinct vectors u and v in Rn
(a) What can you say about the range of the linear operator T? Give an example that illustrates your conclusion. (b) What can you say about the number of vectors that T maps into 0?
Show that the range of the linear operator defined by the equations w1 = 4x1 - 2x2 w2 = 2x1 - x2 is not all of R2, and find a vector that is not in the range.
Determine whether the linear operator T: R2 → R2 defined by the equations is one-to-one; if so, find the standard matrix for the inverse operator, and find T-1 (w1, w2). (a) w1 = x1 + 2x2 w2 = - x1 + x2 (b) w1 = 4x1 - 6x2 w2 = - 2x1 + 3x2
In Exercises use Theorem 4.3.2 to determine whether T: R2 → R2 is a linear operator. (a) T(x, y) = (2x + y, x - y) (b) T(x, y) = (x + 1, y)
Identify the operations on polynomials that correspond to the following operations on vectors. Give the resulting polynomial.(a)(b) (c) (d)
(a) Find a quadratic interpolant to the data (- 2, 1), (0, 1), (1, 4) using the Vandermonde system approach from 1. (b) Repeat using the Newton approach from 4.
(a) Find a polynomial interpolant to the data (- 2, - 10), (- 1, 2), (1, 2), (2, 14) using the Vandermonde system approach from 1. (b) Repeat using the Newton approach from 4. (c) Use 5 to get your answer in part (a) from your answer in part (b). (d) Use 5 to get your answer in part (b) from your
(a) What form does 5 take for lines? (b) What form does 5 take for quadratics? (c) What form does 5 take for quartics?
(a) What matrix corresponds to second differentiation of functions from P2 (giving functions in P0)? (b) What matrix corresponds to second differentiation of functions from P3 (giving functions in P1? (c) Is the matrix for second differentiation the square of the matrix for (first) differentiation?
The third major technique for polynomial interpolation is interpolation using Lagrange interpolating polynomials. Given a set of distinct x-values x0, x1, ... xn, define the n + 1 Lagrange interpolating polynomials for these values by (for i = 0, 1, ... n)Li (x) is a polynomial of exact degree n
The norm of a linear transformation TA: Rn †’ Rn can be defined bywhere the maximum is taken over all nonzero x in Rn. (The subscript indicates that the norm of the linear transformation on the left is found using the Euclidean vector norm on the right.) It is a fact that the largest
(a) Consider the transformation of ax2 + bx + c in P2 to |a| in P0. Show that it does not correspond to a linear transformation by showing that there is no matrix that maps (a, b, c) in R3 to |a| in R. (b) Does the transformation of ax2 + bx + c in P2 to a in p0 correspond to a linear
What matrix corresponds to differentiation in each case? D: P3 → P2
Consider the following matrices. What is the corresponding transformation on polynomials? Indicate the domain pi and the codomain Pj.(a)(b) (c) (d) (e) [0 1 0]
Consider the space of all functions of the form a + bt + cet + de-t, where a, b, c, d are scalars. (a) What function in the space corresponds to the sum of (1, 2, 3, 4) and (- 1, - 2, 0, - 1), assuming that we represent a function in this space as the vector (a, b, c, d)? (b) Is cosh(t) in this
The set of all real-valued functions f defined everywhere on the real line and such that f (1) = 0, with the operations defined in Example 4. In Exercise a set of objects is given, together with operations of addition and scalar multiplication. Determine which sets are vector spaces under the given
The set of all positive real numbers with the operations x + y = xy and kx = xk In Exercise a set of objects is given, together with operations of addition and scalar multiplication. Determine which sets are vector spaces under the given operations. For those that are not vector spaces, list all
Show that the following sets with the given operations fail to be vector spaces by identifying all axioms that fail to hold. (a) The set of all triples of real numbers with the standard vector addition but with scalar multiplication defined by k(x, y, z) = (k2x, k2y, k2z). (b) The set of all
(a) Show that the set of all points in R2 lying on a line is a vector space, with respect to the standard operations of vector addition and scalar multiplication, exactly when the line passes through the origin. (b) Show that the set of all points in R3 lying on a plane is a vector space, with
Do you think that it is possible for a vector u in a vector space to have two different negatives? That is, is it possible to have two different vectors (- u)1 and (-u)2, both of which satisfy Axiom 5? Explain your reasoning.
Use Theorem 5.2.1 to determine which of the following are subspaces of R3. (a) All vectors of the form (a, 0, 0) (b) All vectors of the form (a, 1, 1) (c) All vectors of the form (a, b, c), where b = a + c
In each part, determine whether the given vectors span R3. (a) v1 = (2, 2, 2), v2 = (0, 0, 3), v3 = (0, 1, 1) (b) v1 = (3, 1, 4), v2 = (2, -3, 5), v3 = (5, -2, 9), v4(l, 4, - 1)
Determine whether the following polynomials span P2. p1 = 1 - x + 2x2, p2 = 3 + x, p3 = 5 - x + 4x2, p4 = -2 - 2x + 2x2
Find an equation for the plane spanned by the vectors u = (-1, 1, 1) and v = (3, 4, 4).
Use Theorem 5.2.4 to show that v1 = (1, 6, 4), v2 = (2, 4, -1), v3 = (-1, 2, 5), and w1 = (1, -2, -5), w2 = (0, 8, 9) span the same sub space of R3. Theorem 5.2.4 If S = {v1, v2,..., vr} and S' = {w1, w2,..., wk} are two sets of vectors in a vector space V, then span {v1, v2,..., vk} = span{w1,
Show that the following sets of functions are subspaces of F (-∞, ∞) (a) All everywhere continuous functions (b) All everywhere continuous functions that satisfy f′ + 2f = 0
Indicate whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. (a) If Ax = b is any consistent linar system of m equations in n unknowns, then the solution set is a subspace of Rn. (b) If W is a set of one or more vectors
Use Theorem 5.2.1 to determine which of the following are subspaces of P3. (a) All polynomials a0 + a1x + a2x2 + a3x3 for which a0 = 0 (b) All polynomials a0 + a1x + a2x2 + a3x3 for which a0, a1, a2 and a3 and are integers
Use Theorem 5.2.1 to determine which of the following are subspaces of Mnn. (a) All n x n matrices A such that AT = -A (b) All n x n matrices A such that the linear system Ax = 0 has only the trivial solution
Which of the following are linear combinations of u = (0, -2, 2) and v = (1, 3, -1)? (a) (2, 2, 2) (b) (0, 4, 5)
Express the following as linear combinations of p1 = 2 + x + 4x2, p2 = 1 - x + 3x2, and p3 = 3 + 2x + 5x2. (a) -9 - 7x - 15x2 (b) 0
Show that if S = (v1, v2,..., vr} is a linearly independent set of vectors, then so is every nonempty subset of S.
Show that if {v1, v2,..., vr} is a linearly dependent set of vectors in a vector space V, and if vr+1,..., vn are any vectors in V, then {v1, v2,..., vr, vr+1,..., vn} is also linearly dependent.
Show that if {v1, v2} is linearly independent and V3 does not he in span (v1, v2}, then {v1, v2, v3} is linearly independent.
Use the Wronskian to show that the following sets of vectors are linearly independent. (a) 1, x, ex (b) sin x, cos x, x sin x
Which of the following sets of vectors in R4 are linearly dependent? (a) (3, 8, 7, -3), (1, 5, 3, -1), (2, -1, 2, 6), (1, 4, 0, 3) (b) (0, 0, 2, 2), (3, 3, 0, 0), (1, 1, 0, -1)

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