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Linear Algebra A Modern Introduction 4th edition David Poole - Solutions

In Exercises 1 and 2, find the distance from the point Q to the plane P. 1. Q = (2, 2, 2), P with equation x + y - z = 0 2. Q = (O, 0, O), P with equation x - 2y + 2z = 1
In Exercises 1-4, write the equation of the line passing through P with direction vector d in (a) vector form and (b) parametric form1.2. 3. 4.
Find the point R on f that is closest to Q in Exercise 27.In exercise 27
In Exercises 1 and 2, find the distance between the parallel lines.1.2.
In Exercises 1 and 2, find the distance between the parallel planes. 1. 2x + y - 2z = 0 and 2x + y - 2z = 5 2. x + y + z = 1 and x + y + z = 3
Prove that, in R2, the distance between parallel lines with equations n · x = c1 and n · x = c2 is given by |c1 - c2| / ||n||.
Prove that the distance between parallel planes with equations n · x = d1 and n · x = d2 is given by |d1 - d2| / ||n||.
In Exercises 1-2, find the acute angle between the planes with the given equations. 1. x + y + z = 0 and 2x + y - 2z = 0 2. 3x - y + 2z = 5 and x + 4y - z = 2
The plane given by x + y + 2z = 0 and the line given by x = 2 + t y = 1 - 2t z = 3 + t
The plane given by 4x - y - z given by x = t y = 1 + 2t z = 2 + 3t
Using the fact that n is orthogonal to every vector in P (and hence to p), solve for c and thereby find an expression for p in terms of v and n.
Use the method of Exercise 43 to find the projection of onto the planes with the following equations:(a) x + y + z = 0 (b) 3x - y + z = 0 (c) x - 2z = 0 (d) 2x - 3y + z = 0
In Exercises 1 and 2, write the equation of the plane passing through P with normal vector n in (a) normal form and (b) general form.1.2.
In Exercises 1 and 2, write the equation of the plane passing through P with direction vectors u and v in (a) vector form and (b) parametric form.1.2.
f1 acting due north with a magnitude of 12N and f2 acting due east with a magnitude of 5N.
A lawn mower has a mass of 30 kg. It is being pushed with a force of 100 N. If the handle of the lawn mower makes an angle of 45° with the ground, what is the horizontal component of the force that is causing the mower to move forward?
A sign hanging outside Joe's Diner has a mass of 50 kg (Figure 1 .82). If the supporting cable makes an angle of 60° with the wall of the building, determine the tension in the cable.
A sign hanging in the window of Joe's Diner has a mass of 1 kg. If the supporting strings each make an angle of 45° with the sign and the supporting hooks are at the same height (Figure 1 .83), find the tension in each string.
A painting with a mass of 15 kg is suspended by two wires from hooks on a ceiling. If the wires have lengths of 15 cm and 20 cm and the distance between the hooks is 25 cm, find the tension in each wire.
A painting with a mass of 20 kg is suspended by two wires from a ceiling. If the wires make angles of 30° and 45° with the ceiling, find the tension in each wire.
f1 acting due west with a magnitude of 15N and f2 acting due south with a magnitude of 20N.
f1 acting with a magnitude of 8N and f2 acting at an angle of 60° to f1 with a magnitude of 8N.
f1 acting with a magnitude of 4N and f2 acting at an angle of 135° to f1 with a magnitude of 6N
f1 acting due east with a magnitude of 2N, f2 acting due west with a magnitude of 6 N, and f3 acting at an angle of 60° to f1 with a magnitude of 4 N
f1 acting due east with a magnitude of 10N, f2 acting due north with a magnitude of 13 N, f3 acting due west with a magnitude of 5N, and f4 acting due south with a magnitude of 8N
Resolve a force of 10 N into two forces perpendicular to each other so that one component makes an angle of 60° with the 10 N force.
A 10 kg block lies on a ramp that is inclined at an angle of 30° (Figure 1 .80). Assuming there is no friction, what force, parallel to the ramp, must be applied to keep the block from sliding down the ramp?
A tow truck is towing a car. The tension in the tow cable is 1 500 N and the cable makes a 45° with the horizontal, as shown in Figure 1. 81. What is the vertical force that tends to lift the car off the ground?
Mark each of the following statements true or false: (a) For vectors u, v, and w in Rn, if u + w = v + w, then u = v. (b) For vectors u, v, and w in Rn, if u ∙ w = v ∙ w, then u = v. (c) For vectors u, v, and w in R3, if u is orthogonal to v, and v is orthogonal to w, then u is orthogonal to
Find the general equation of the plane through the points A(1, 1, 0), B (1, 0, 1), and C(0, 1, 2).
Find the area of the triangle with vertices A(1, 1, 0), B (1 , 0, 1) , and C(0, 1, 2).
Find the midpoint of the line segment between A = (5, 1, - 2) and B = (3, - 7, 0).
Why are there no vectors u and v in Rn such that ||u|| = 2, ||v|| = 3, and u · v = -7?
Find the distance from the point (3, 2, 5) to the plane whose general equation is 2x + 3y - z = 0.
Find the distance from the point (3, 2, 5) to the line with parametric equations x = t, y = 1 + t, z = 2 + t.
Compute 3 - (2 + 4)3 (4 + 3)2 in Z5.
If possible, solve 3 (x + 2) = 5 in Z7.
Compute [2, 1, 3, 3] · [3, 4, 4, 2] in Z45.
Ifand the vector 4u + v is drawn with its tail at the point (10, - 10), find the coordinates of the point at the head of 4u + v.
Let u = [l, 1, 1, 0] in Z42. How many binary vectors v satisfy u ∙ v = 0?
IfAnd 2x + u = 3(x - v), solve for x.
Find the angle between the vectors [- 1, 1, 2] and [2, 1, - 1].
Find the projection of
Find a unit vector in the xy-plane that is orthogonal to
Find the general equation of the plane through the point (1, 1, 1) that is perpendicular to the line with parametric equations x = 2 - t y = 3 + 2 t z = - 1 + t
Find the general equation of the plane through the point (3, 2, 5) that is parallel to the plane whose general equation is 2x + 3y - z = 0.
In Exercises 1-6, determine which equations are linear equations in the variables x, y, and z. If any equation is not linear, explain why not. 1. x - πy + 3√5z = 0 2. x2 + y2 + z2 = 1 3. x-1 + 7y + z = sin (π/9)
In Exercises 1-3, find the solution set of each equation. 1. 3x - 6y = 0 2. 2x1 + 3x2 = 5 3. x + 2y + 3z = 4
In Exercises 1-3, draw graphs corresponding to the given linear systems. Determine geometrically whether each system has a unique solution, infinitely many solutions, or no solution. Then solve each system algebraically to confirm your answer. 1. x + y = 0 2x + y = 3 2. x - 2y = 7 3x + y = 7 3. 3x
In Exercises 1-3, solve the given system by back substitution. 1. x - 2y = 1 y = 3 2. 2u - 3v = 5 2v = 6 3. x - y + z = 0 2y - z = 1 3z = - 1
The systems in Exercises 1 and 2 exhibit a "lower triangular" pattern that makes them easy to solve by forward substitution. (We will encounter forward substitution again in Chapter 3.) Solve these systems. 1. x = 2 2x + y = -3 -3x -4y + z = -10 2. x1 = -1 -1/2 x1 + x2 = 5 3/2 x1 + 2x2 + x3 = 7
Find the augmented matrices of the linear systems in Exercises 1-3. 1. x - y = 0 2x + y = 3 2. 2x1 + 3x2 - x3 = 1 x1 + x3 = 0 -x1 + 2x2 - 2x3 = 0 3. x + 5y = - 1 -x + y = - 5 2x + 4y = 4
In Exercises 1 and 2, find a system of linear equations that has the given matrix as its augmented matrix.1.2.
(a) Find a system of two linear equations in the variables x and y whose solution set is given by the parametric equations x = t and y = 3 - 2t. (b) Find another parametric solution to the system in part (a) in which the parameter is s and y = s.
(a) Find a system of two linear equations in the variables x1, x2, and x3 whose solution set is given by the parametric equations x1 = t, x2 = 1 + t, and x3 = 2 - t. (b) Find another parametric solution to the system in part (a) in which the parameter is s and x3 = s.
In Exercises 1-3, find a linear equation that has the same solution set as the given equation (possibly with some restrictions on the variables). 1. 2x + y = 7 - 3y 2. x2 - y2 / x - y = 1 3. 1/ x + 1/ y = 4/xy
In Exercises 1-3, determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form.1.2. 3.
Reverse the elementary row operations used in Example 2.9 to show that we can convertinto
In Exercises 1 and 2, show that the given matrices are row equivalent and find a sequence of elementary row operations that will convert A into B.1.2.
What is wrong with the following "proof" that every matrix with at least two rows is row equivalent to a matrix with a zero row? Perform R2 + R1 and R1 + R2. Now rows 1 and 2 are identical. Now perform R2 - R1 to obtain a row of zeros in the second row.
What is the net effect of performing the following sequence of elementary row operations on a matrix (with at least two rows)? R2 + R1, R1 - R2, R2 + R1, - R1
Students frequently perform the following type of calculation to introduce a zero into a matrix:However, 3R2 - 2R1 is not an elementary row operation. Why not? Show how to achieve the same result using elementary row operations.
Consider the matrixShow that any of the three types of elementary row operations can be used to create a leading 1 at the top of the first column. Which do you prefer and why?
What are the possible reduced row echelon forms of 3 X 3 matrices?
In Exercises 1-3, solve the given system of equations using either Gaussian or Gauss-Jordan elimination. 1. X1 + 2X2 - 3X3 = 9 2X1 - X2 + X3 = 0 4x1 - X2 + x3 = 4 2. x - y + z = 0 -x + 3y + z = 5 3x + y + 7z = 2 3. X1 - 3X2 - 2X3 = 0 -X1 + 2X2 + X3 = 0 2x1 + 4x2 + 6x3 = 0
In Exercises 1-3, determine by inspection (i.e., without performing any calculations) whether a linear system with the given augmented matrix has a unique solution, infinitely many solutions, or no solution. Justify your answers. 1. 2. 3.
Show that if ad - be -=!= 0, then the system ax + by = r cx + dy = s
In Exercises 1-3, for what value(s) of k, if any, will the systems have (a) no solution, (b) a unique solution, and (c) infinitely many solutions? 1. kx + 2y = 3 2x - 4y = -6 2. x + ky = 1 kx + y = 1 3. x - 2y + 3z = 2 x + y + z = k 2x - y + 4z = k2
Give examples of homogeneous systems of m linear equations in n variables with m = n and with m > n that have
In Exercises 1 and 2, find the line of intersection of the given planes. 1. 3x + 2y + z = - 1 and 2x - y + 4z = 5 2. 4x + y + z = 0 and 2x - y + 3z = 2
(a) Give an example of three planes that have a common line of intersection (Figure 2.4).Figure 2.4 (b) Give an example of three planes that intersect in pairs but have no common point of intersection (Figure 2.5). Figure 2.5 (c) Give an example of three planes, exactly two of which are parallel
In Exercises 1 and 2, determine whether the lines x = p + su and x = q + tv intersect and, if they do, find their point of intersection.1.2.
50.Describe all points Q = (a, b, c) such that the line through Q with direction vector v intersects the line with equation x = p + su.
Recall that the cross product of vectors u and v is a vector u ( v that is orthogonal to both u and v. (See Exploration: The Cross Product in Chapter 1.) Ifshow that there are infinitely many vectors that simultaneously satisfy u· x = 0 and v · x = 0 and that all are multiples of
Show that the lines x = p + su and x = q + tv are skew lines. Find vector equations of a pair of parallel planes, one containing each line.
In Exercises 1-3, solve the systems of linear equations over the indicated 1. x + 2y = 1 over x + y = 22. x + y = 1 over y + z = Ox + z = 13. x + y = 1 overy + z = Ox + z = 1
Prove the following corollary to the Rank Theorem:Let A be an m X n matrix with entries in
When p is not prime, extra care is needed in solving a linear system (or, indeed, any equation) over2x + 3y = 44x + 3y = 2
In Exercises 1-3, use elementary row operations to reduce the given matrix to (a) row echelon form and (b) reduced row echelon form.1.2. 3.
In Exercises 1-3, determine if the vector v is a linear combination of the remaining vectors.1.2. 3.
In Exercises 1-3, describe the span of the given vectors (a) geometrically and (b) algebraically.1.2. 3.
The general equation of the plane that contains the points (1, 0, 3), (-1, 1, -3), and the origin is of the form ax + by + cz = 0. Solve for a, b, and c.
Prove that u, v, and w are all in span (u, v, w).
Prove that u, v, and w are all in span (u, u + v, u + v + w).
(a) Prove that if u1, ( ( ( ( um are vectors in , S = {u1, u2, ( ( ( ( uk}, and T = {u1( ( ( ( ( uk, uk+1, ( ( ( ( um}, then span (S) ( span (T). [Hint: Rephrase this question in terms of linear combinations.](b) Deduce that if = span (S), then = span (T) also.
(a) Suppose that vector w is a linear combination of vectors u1( ( ( ( ( uk and that each u; is a linear combination of vectors v1( ( ( ( ( vm. Prove that w is a linear combination ofv1( ( ( ( ( vm and therefore span (u1( ( ( ( ( uk) ( span (v1( ( ( ( ( vm).(b) In part (a), suppose in addition
Use the method of Example 2.23 and Theorem 2.6 to determine if the sets of vectors in Exercises 1-3 are linearly independent. If, for any of these, the answer can be determined by inspection (i.e., without calculation), state why. For any sets that are linearly dependent, find a dependence
(a) If the columns of an n X n matrix A are linearly independent as vectors in, what is the rank of A? Explain.(b) If the rows of an n X n matrix A are linearly independent as vectors in, what is the rank of A? Explain.
(a) If vectors u, v, and w are linearly independent, will u + v, v + w, and u + w also be linearly independent? Justify your answer. (b) If vectors u, v, and w are linearly independent, will u - v, v - w, and u - w also be linearly independent? Justify your answer.
Prove that two vectors are linearly dependent if and only if one is a scalar multiple of the other. [Hint: Separately consider the case where one of the vectors is O.]
Give a "row vector proof" of Theorem 2.8.
Prove that every subset of a linearly independent set is linearly independent.
Suppose that S = {v1( ( ( ( ( vk, v} is a set of vectors in some and that v is a linear combination of v1( ( ( ( ( vk. If S' = {v1( ( ( ( ( vk, prove that span (S) = span (S'). [Hint: Exercise 2 l (b) is helpful here.]
Let {v1( ( ( ( ( vk be a linearly independent set of vectors in, and let v be a vector in. Suppose that v = c1v1 + c2v2 + ( ( ( + ck vk with c1 ‰ 0. Prove that {v, v2( ( ( ( ( vk} is linearly independent.
In Exercises 1 and 2, determine if the vector b is in the span of the columns of the matrix A.1.2.
Suppose that, in Example 2.27, 400 units of food A, 600 units of B, and 600 units of C are placed in the test tube each day and the data on daily food consumption by the bacteria (in units per day) are as shown in Table 2.6. How many bacteria of each strain can coexist in the test tube and consume
Figure 2 . 1 8 shows a network of water pipes with flows measured in liters per minute.(a) Set up and solve a system of linear equations to find the possible flows.(b) If the flow through AB is restricted to 5 L/min, what will the flows through the other two branches be?(c) What are the minimum and
The downtown core of Gotham City consists of one-way streets, and the traffic flow has been measured at each intersection. For the city block shown in Figure 2.19, the numbers represent the average numbers of vehicles per minute entering and leaving intersections A, B, C, and D during business
A network of irrigation ditches is shown in Figure 2.20, with flows measured in thousands of liters per day.(a) Set up and solve a system of linear equations to find the possible flows f1( ( ( ( ( f5. (b) Suppose DC is closed. What range of flow will need to be maintained through DB?
(a) Set up and solve a system of linear equations to find the possible flows in the network shown in Figure 2.2 1.

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