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mathematics
linear algebra

Linear Algebra and Its Applications 4th edition David C. Lay - Solutions

Determine by inspection whether the vectors in Exercises 1-2 are linearly independent. Justify each answer.1.2.
How many pivot columns must a 6 ( 4 matrix have if its columns are linearly independent? Why?
How many pivot columns must a 4 ( 6 matrix have if its columns span? Why?
Construct 3 ( 2 matrices A and B such that Ax = 0 has a nontrivial solution, but Bx = 0 has only the trivial solution.
a. Fill in the blank in the following statement: "If A is an m ( n matrix, then the columns of A are linearly independent if and only if A has pivot columns." b. Explain why the statement in (a) is true.
Exercises 1 and 2 should be solved without performing row operations. [Write Ax = 0 asa vector equation.]1. Given
1. If v1, v2, v3 are in and v3 is not a linear combination of v1, v2, then {v1, v2, v3} is linearly independent.2. If v1( ( ( ( ( v4 are in and {v1, v2, v3} is linearly dependent, then {v1, v2, v3, v4} is also linearly dependent.
Suppose A is an m ( n matrix with the property that for all b in the equation Ax = b has at most one solution. Use the definition of linear independence to explain why the columns of A must be linearly independent.
In Exercises 1 and 2, use as many columns of A as possible to construct a matrix B with the property that the equation Bx = 0 has only the trivial solution. Solve Bx = 0 to verify your work. 1. 2.
With A and B as in Exercise 41, select a column v of A that was not used in the construction of B and determine if v is in the set spanned by the columns of B. (Describe your calculations.)
Repeat Exercise 43 with the matrices A and B from Exercise 42. Then give an explanation for what you discover, assuming that B was constructed as specified.
In Exercises 1-2, determine if the columns of the matrix form a linearly independent set. Justify each answer.1.2.
In Exercises 1 and 2, (a) for what values of h is v3 in Span {v1, v2}, and (b) for what values of h is {v1, v2, v3} linearly dependent? Justify each answer.1.2.
In Exercises 1-2, use a rectangular coordinate system to plot1. 2.
Use the fact that T is linear to find the images under T of 2u, 3v, and 2u + 3v.
The figure shows vectors u, v, and w, along with the images T(u) and T(v) under the action of a linear transformation T :. Copy this figure carefully, and draw the image T(w) as accurately as possible. [First, write w as a linear combination of u and v.]
In Exercises 1 and 2, mark each statement True or False. Justify each answer.1. a. A linear transformation is a special type of function.d. Every linear transformation is a matrix transformation. e. A transformation T is linear if and only if T(c1v1 + c2v2) = c1T(v1 + c2T(v2) for all v1 and v2 in
a. Show that ( is a linear transformation when b = 0.b. Find a property of a linear transformation that is violated when b ( 0.c. Why is ( called a linear function?
An affine transformation T: has the form T(x) = Ax + b, with A an m ( n matrix and b in. Show that T is not a linear transformation when b ( 0. (Affine transformations are important in computer graphics.)
Given v ( 0 and p in, the line through p in the direction of v has the parametric equation x = p + t v. Show that a linear transformation T: maps this line onto another line or onto a single point (a degenerate line).
a. Show that the line through vectors p and q in may be written in the parametric form x = (1 -()p + tq. (Refer to the figure with Exercises 21 and 22 in Section 1.5.) b. The line segment from p to q is the set of points of the form (1 - t)p + tq for 0
Let u and v be linearly independent vectors in, and let P be the plane through u, v, and 0. The parametric equation of P is x = su + tv (with s, t in). Show that a linear transformation T: maps P onto a plane through 0, or onto a line through 0, or on to just the origin in. What must be true about
Let u and v be vectors in. It can be shown that the set P of all points in the parallelogram determined by u and v has the form su + bv, for0 ( ( ( 1, 0 ( b (1. Let T: ( be a linear transformation. Explain why the image of a point in P under the transformation T lies in the parallelogram determined
Let T: be the linear transformation that reflects each point through the x2-axis. Make two sketches similar to Fig. 6 that illustrate properties (i) and (ii) of a linear transformation.
In Exercises 1-2, with T defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is unique.1.2.
Suppose vectors v1( ( ( ( ( vp span, and let T: ( be a linear transformation. Suppose T(v;) = 0 for i = 1,..., p. Show that T is the zero transformation. That is, show that if x is any vector in, then T(x) = 0.
Let T:( be a linear transformation, and let {v1, v2, v3} be a linearly dependent set in . Explain why the set {T(v1), T(v2), T(v3)} is linearly dependent.
In Exercises 1 and 2, the given matrix determines a linear transformation T. Find all x such that T(x) = 0.1.2.
Solve each system in Exercises 1-2 by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure described in this section. 1. x1 + 5x2 = 7 -2x1 - 7x2 = -5 2. 3x1 + 6x2 = -3 5x1 + 7x2 = 10
Let A be a 6 ( 5 matrix. What must a and b be in order to define T : ( by T(x) = Ax?
How many rows and columns must a matrix A have in order to define a mapping from into by the rule T(x) = Ax?
1.
In Exercises 1-2, assume that T is a linear transformation. Find the standard matrix of T.
A linear transformation T: first reflects points through the x1 -axis and then reflects points through the x2- axis. Show that T can also be described as a linear transformation that rotates points about the origin. What is the angle of that rotation?
Show that the transformation in Exercise 10 is merely a rotation about the origin. What is the angle of the rotation?
Let T: be the linear transformation such that T(e1) and T(e2) are the vectors shown in the figure. Using the figure, sketch the vector T(2,1).
under the transformation T.
In Exercises 1 and 2, fill in the missing entries of the matrix, assuming that the equation holds for all values of the variables.1.2.
In Exercises 1-2, show that T is a linear transformation by finding a matrix that implements the mapping. x1 ,x2,... are not vectors but are entries in vectors.1.2.
Solve the systems in exercises 1-2. 1. x2 + 5x3 = -4 x1 + 4x2 + 3x3 = -2 2x1 + 7x2 + x3 = -2 2. x1 - 5x2 + 4x3 = -3 2x1 - 7x2 + 3x3 = -2 -2x1 + x2 + 7x3 = -1
Let T : be a linear transformation such that T(x1,x2) = (x1 + x2, 4x1 + 5x2). Find x such that T (x) = (3, 8).
Let T: be a linear transformation with T(x1, x2) = (2x1 - x2, - 3x1 + x2, 2x1 - 3x2). Find x such that T (x) = (0, -1, -4).
In Exercises 1-2, determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer.1. The transformation in Exercise 172. The transformation in Exercise 2
In Exercises 1 and 2, describe the possible echelon forms of the standard matrix for a linear transformation T. Use the notation of Example 1 in Section 1.2.
Let T: Rn ! Rm be a linear transformation, with A its standard matrix. Complete the following statement to make it true: "T is one-to-one if and only if A has pivot columns." Explain why the statement is true
Let T : Rn ! Rm be a linear transformation, with A its standard matrix. Complete the following statement to make it true: "T maps Rn onto Rm if and only if A has pivot columns." Find some theorems that explain why the statement is true.
Verify the uniqueness of A in Theorem 10. Let T: Rn ! Rm be a linear transformation such that T(x) = Bx for some m x n matrix B. Show that if A is the standard matrix for T, then A = B.
Let S : Rp ! Rn and T : Rn ! Rm be linear transformations. Show that the mapping x ! T(S(x)) is a linear transformation (from Rp toRm).
If a linear transformation T : Rn ! Rm maps Rn onto Rm, can you give a relation between m and n? If T is one-to-one, what can you say about m and n?
1.
Determine if the systems in Exercises 1 and 2 are consistent. Do not completely solve the systems. 1. x1 - 6x2 = 5 x2 - 4x3 + x4 = 0 -x1 + 6x2 + x3 + 5x4 = 3 -x2 + 5x3 + 4x4 = 0 2. 2x1 - 4x4 = -10 3x2 + 3x3 = 0 x3 + 4x4 = -1 -3x1 + 2x2 + 3x3 + x4 = 5
Do the three lines 2x1 + 3x2 = -1, 6x1 + 5x2 = 0, and 2x1 - 5x2 = 7 have a common point of intersection? Explain.
Do the three planes 2x1 + 4x2 + 4x3 = 4, x2 -2x3 = -2, and 2x1 + 3x2 = 0 have at least one common point of intersection? Explain.
In Exercises 19-22, determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.1.2.
In Exercises 23 and 24, key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer. (If true, give the approximate location where a similar
Find an equation involving g, h, and k that makes this augmented matrix correspond to a consistent system:
Suppose the system below is consistent for all possible values of f and g. What can you say about the coefficients c and d? Justify your answer.2x1 + 4x2 = (cx1 + dx2 = g
Suppose a, b, c, and d are constants such that a is not zero and the system below is consistent for all possible values of f and g. What can you say about the numbers a, b, c, and d? Justify your answer. (x1 + bx2 = ( cx1 + dx2 = g
Construct three different augmented matrices for linear systems whose solution set is x1 = 3, x2 = -2, x3 = -1.
Write a system of four equations whose solution gives estimates for the temperatures T1( ( ( ( (T4.
Solve the system of equations from Exercise 33. [To speed up the calculations, interchange rows 1 and 4 before starting "replace" operations.]
Consider each matrix in Exercises 1 and 2 as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system.1.2.
In Exercises 1-2, the augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system. 1. 2.
The container of a breakfast cereal usually lists the number of calories and the amounts of protein, carbohydrate, and fat contained in one serving of the cereal. The amounts for two common cereals are given below. Suppose a mixture of these two cereals is to be prepared that contains exactly 295
In a certain region, about 6% of a city's population moves to the surrounding suburbs each year, and about 4% of the suburban population moves into the city. In 2010, there were 10,000,000 residents in the city and 800,000 in the suburbs. Set up a difference equation that describes this situation,
In 1994, the population of California was 31,524,000, and the population living in the United States but outside California was 228,680,000. During the year, it is estimated that 516,100 persons moved from California to elsewhere in the United States, while 381,262 persons moved to California from
Budget® Rent A Car in Wichita, Kansas has a fleet of about 500 cars, at three locations. A car rented at one location may be returned to any of the three locations. The various fractions of cars returned to the three locations are shown in the matrix below. Suppose that on Monday there are 295
Let M and xo be as in Example 3. a. Compute the population vectors xi, for k = 1 20. Discuss what you find. b. Repeat part (a) with an initial population of 350,000 in the city and 650,000 in the suburbs. What do you find?
Study how changes in boundary temperatures on a steel plate affect the temperatures at interior points on the plate. a. Begin by estimating the temperatures T1, T2, T3, T4 at each of the sets of four points on the steel plate shown in the figure. In each case, the value of Tk is approximated by
One serving of Shredded Wheat supplies 160 calories, 5 g of protein, 6 g of fiber, and 1 g of fat. One serving of Crispix® supplies 110 calories, 2 g of protein, .1 g of fiber, and .4 g of fat. a. Set up a matrix B and a vector n such that Bu gives the amounts of calories, protein, fiber, and fat
After taking a nutrition class, a big Annie's® Mac and Cheese fan decides to improve the levels of protein and fiber in her favorite lunch by adding broccoli and canned chicken. The nutritional information for the foods referred to in this exercise are given in the table below.a. If she wants
The Cambridge Diet supplies .8 g of calcium per day, in addition to the nutrients listed in the Table 1 for Example 1. The amounts of calcium per unit (100 g) supplied by the three ingredients in the Cambridge Diet are as follows: 1.26 g from nonfat milk, .19 g from soy flour, and .8 g from whey.
In Exercises 1-2, write a matrix equation that determines the loop currents. [NI] If MATLAB or another matrix program is available, solve the system for the loop currents.1.2.
In a certain region, about 7% of a city's population moves to the surrounding suburbs each year, and about 5% of the suburban population moves into the city. In 2010, there were 800,000 residents in the city and 500,000 in the suburbs. Set up a difference equation that describes this situation,
Exercises 1 and 2 use the notation of Example 1 for matrices in echelon form. Suppose each matrix represents the augmented matrix for a system of linear equations. In each case, determine if the system is consistent. If the system is consistent, determine if the solution is unique. 1. a. b. 2.
In Exercises 1 and 2, determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.1.2.
In Exercises 1 and 2, choose h and k such that the system has (a) no solution, (b) a unique solution, and (c) many solutions. Give separate answers for each part. 1. x1 + hx2 = 2 4x1 C 8x2 = k 2. x1 - 3x2 = 1 2x1 + hx2 = k
In Exercises 1 and 2, mark each statement True or False. Justify each answer. 1. a. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations. b. The row reduction algorithm applies only to augmented matrices for a
Suppose the coefficient matrix of a linear system of four equations in four variables has a pivot in each column. Explain why the system has a unique solution.
Suppose a system of linear equations has a 3 ( 5 augmented matrix whose fifth column is not a pivot column. Is the system consistent? Why (or why not)?
Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.
Suppose a 3 ( 5 coefficient matrix for a system has three pivot columns. Is the system consistent? Why or why not?
Restate the last sentence in Theorem 2 using the concept of pivot columns: "If a linear system is consistent, then the solution is unique if and only if _____________"
A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system. Can such a system have a unique solution? Explain.
Row reduce the matrices in Exercises 1 and 2 to reduced echelon form. Circle the pivot positions in the final matrix and in the original matrix, and list the pivot columns.1.2.
1. Give an example of an inconsistent underdetermined system of two equations in three unknowns. 2. A system of linear equations with more equations than unknowns is sometimes called an over determined system. Can such a system be consistent? Illustrate your answer with a specific system of three
Suppose an n ( (n + 1) matrix is row reduced to reduced echelon form. Approximately what fraction of the total number of operations (flops) is involved in the backward phase of the reduction when n = 20? when n = 200? Suppose experimental data are represented by a set of points in the plane. An
Find the interpolating polynomial p(t) = (0 + (1t + a2t2 for the data (1,6), (2,15), (3, 28). That is, find (0, (1; and (2 such that
In a wind tunnel experiment, the force on a projectile due to air resistance was measured at different velocities:Find an interpolating polynomial for these data and estimate the force on the projectile when the projectile is traveling at 750 ft/sec. Use p(t) = a0 + a1t + a2t2 + a3t3 + a4t4 C a5t5.
1. Describe the possible echelon forms of a nonzero 2 x 2 matrix. Use the symbols ■, *, and 0, as in the first part of Example 1. 2. Repeat Exercise 5 for a nonzero 3 x 2 matrix.
Find the general solutions of the systems whose augmented ma trices are given in Exercises 1-2. 1. 2.
In Exercises 1 and 2, compute u + v and u - 2v.1.2.
In Exercises 1 and 2, determine if b is a linear combination of a1, a2, and a3.1.2.
In Exercises 1 and 2, determine if b is a linear combination of the vectors formed from the columns of the matrix A.1.2.
For what value(s) of h is b in the plane spanned by a1 and a2?Let
For what value(s) of h is y in the plane generated by v1 and v2?Let
In Exercises 1 and 2, list five vectors in Span {v1, v2}. For each vector, show the weights on v1 and v2 used to generate the vector and list the three entries of the vector. Do not make a sketch.1.2.
Give a geometric description of Span{v1,v2}. For the vectors
Span {u, v} for all h and k.
In Exercises 1 and 2, mark each statement True or False. Justify each answer.1. a Another notation for the vectorb. The points in the plane corresponding to And Lie on a line through the origin. c. An example of a linear combination of vectors v1 and v2 is the vector 1/2 v1.

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