Let T : ?3? ?3 be the linear transformation that reflects

Problem 25E A large apartment building is to be built using modular construction techniques. The arrangement of apartments on any particular floor is to be chosen from one of three basic floor plans. Plan A has 18 apartments on one floor, including 3 three-bedroom units, 7 two-bedroom units, and 8 one-bedroom units. Each floor of plan B includes 4 three bedroom units, 4 two-bedroom units, and 8 one-bedroom units. Each floor of plan C includes 5 three-bedroom units, 3 two-bedroom units, and 9 one-bedroom units. Suppose the building contains a total of x1 floors of plan A, x2 floors of plan B, and x3 floors of plan C. a. What interpretation can be given to the vector b. Write a formal linear combination of vectors that expresses the total numbers of three-, two-, and one bedroom apartments contained in the building. c. [M] Is it possible to design the building with exactly 66 three-bedroom units, 74 two-bedroom units, and 136 one bedroom units? If so, is there more than one way to do it? Explain your answer.

Determine h and k such that the solution set of the system (i) is empty, (ii) contains a unique solution, and (iii) contains infinitely many solutions. a. \(x_{1}+3 x_{2}=k\) \(4 x_{1}+h x_{2}=8\) b. \(-2 x_{1}+h x_{2}=1\) \(6 x_{1}+k x_{2}=-2\)

Problem 1E Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case. a. Every matrix is row equivalent to a unique matrix in echelon form. b. Any system of n linear equations in n variables has at most n solutions. c. If a system of linear equations has two different solutions, it must have infinitely many solutions. d. If a system of linear equations has no free variables, then it has a unique solution. e. If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets. f. If a system Ax = b has more than one solution, then so does the system Ax = 0. g. If A is an m × n matrix and the equation Ax = b is consistent for some b, then the columns of A span ?m. h. If an augmented matrix [A b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax = b is consistent. i. If matrices A and B are row equivalent, they have the same reduced echelon form. j. The equation Ax = 0 has the trivial solution if and only if there are no free variables. k. If A is an m × n matrix and the equation Ax = b is consistent for every b in ?m, then A has m pivot columns. l. If an m × n matrix A has a pivot position in every row, then the equation Ax = b has a unique solution for each b in ?m. m. If an n × n matrix A has n pivot positions, then the reduced echelon form of A is the n × n identity matrix. n. If 3 × 3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations. o. If A is an m × n matrix, if the equation Ax = b has at least two different solutions, and if the equation Ax = c is consistent, then the equation Ax = c has many solutions. p. If A and B are row equivalent m × n matrices and if the columns of A span ?m, then so do the columns of B. q. If none of the vectors in the set S = {v1, v2, v3} in ?3 is a multiple of one of the other vectors, then S is linearly independent. r. If {u, v, w} is linearly independent, then u, v, and w are not in ?2. s. In some cases, it is possible for four vectors to span ?m. t. If u and v are in ?m, then –u is in Span {u, v} u. If u, v, and w are nonzero vectors in ?2, then w is a linear combination of u and v. v. If w is a linear combination of u and v in T : ?n, then u is a linear combination of v and w. w. Suppose that v1, v2 and v3 are in ?5, v2 is not a multiple of v1, and v3 is not a linear combination of v1 and v2. Then {v1, v2, v3} is linearly independent. x. A linear transformation is a function. y. If A is a 6 × 5 matrix, the linear transformation x ? Ax cannot map ?5 onto ?6. z. If A is an m × n matrix with m pivot columns, then the linear transformation x ? Ax is a one-to-one mapping.

Problem 4E Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot position in each column. Explain why the system has a unique solution.

Let a and b represent real numbers. Describe the possible solution sets of the (linear) equation ax = b. [Hint: The number of solutions depends upon a and b.]

Consider the problem of determining whether the following system of equations is consistent: \(4 x_{1}-2 x_{2}+7 x_{3}=-5\) \(8 x_{1}-3 x_{2}+10 x_{3}=-3\) a. Define appropriate vectors, and restate the problem in terms of linear combinations. Then solve that problem. b. Define an appropriate matrix, and restate the problem using the phrase “columns of A.” c. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.

Problem 3E The solutions (x, y, z) of a single linear equation ax + by + cz = d form a plane in ?3& when a, b, and c are not all zero. Construct sets of three linear equations whose graphs (a) intersect in a single line, (b) intersect in a single point, and (c) have no points in common. Typical graphs are illustrated in the figure.

Consider the problem of determining whether the following system of equations is consistent for all \(b_1, b_2, b_3\): \(\begin{aligned} 2 x_{1}-4 x_{2}-2 x_{3} &=b_{1} \\ -5 x_{1}+x_{2}+x_{3} &=b_{2} \\ 7 x_{1}-5 x_{2}-3 x_{3} &=b_{3} \end{aligned}\) a. Define appropriate vectors, and restate the problem in terms of Span {\(v_1, v_2, v_3\)}. Then solve that problem. b. Define an appropriate matrix, and restate the problem using the phrase “columns of A.” c. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.

Problem 9E Write the vector as the sum of two vectors, one on the line and one on the line .

Problem 10E 10. Let a1, a2, and b be the vectors in ?2 shown in the figure, and let A = [a1 a2]. Does the equation Ax = b have a solution? If so, is the solution unique? Explain.

Describe the possible echelon forms of the matrix A. Use the notation of Example 1 in Section 1.2. a. A is a \(2 \times 3\) matrix whose columns span \(\mathbb{R}^{2}\). b. A is a \(3 \times 3\) matrix whose columns span \(\mathbb{R}^{3}\).

Write the reduced echelon form of a \(3 \times 3\) matrix A such that the first two columns of A are pivot columns and \(A\left[\begin{array}{r}3 \\ -2 \\ 1\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]\)

Construct a 2 X 3 matrix A, not in echelon form, such that the solution of Ax = 0 is a plane in \(\mathbb{R}^{3}\).

Problem 11E Construct a 2 × 3 matrix A, not in echelon form, such that the solution of Ax = 0 is a line in ?3.

Problem 14E Determine the value(s) of a such that Is linearly independent.

Problem 15E In (a) and (b), suppose the vectors are linearly independent. What can you say about the numbers a,…, f? Justify your answers. [Hint: Use a theorem for (b).]

Problem 16E Use Theorem 7 in Section 1.7 to explain why the columns of the matrix A are linearly independent. Theorem 7: Characterization of Linearly Dependent Sets An indexed set S = {v1,…, vp} of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. In fact, if S is linearly dependent and v1 ? 0, then some vj (with j > 1) is a linear combination of the preceding vectors, v1,…, vj–1.

Problem 21E Let T : ?3? ?3 be the linear transformation that reflects each vector through the plane x2 = 0. That is, T = (x1, x2, x3) = (x1, –x2, x3). Find the standard matrix of T.

Problem 18E Suppose {v1, v2} is a linearly independent set in ?n. Show that {v1, v1 + v2} is also linearly independent.

Problem 17E Explain why a set {v1, v2, v3, v4} in ?5 must be linearly independent when {v1, v2, v3} is linearly independent and v4 is not in Span {v1, v2, v3}.

Problem 19E Suppose v1, v2, v3 are distinct points on one line in ?3. The line need not pass through the origin. Show that {v1, v2, v3} is linearly dependent.

Let \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) be a linear transformation, and suppose T (u) = v. Show that T (–u) = –v.

Problem 24E The following equation describes a Givens rotation in ?3. Find a and b.

Problem 23E A Givens rotation is a linear transformation from ?n& to ?n& used in computer programs to create a zero entry in a vector (usually a column of a matrix). The standard matrix of a Givens rotation in ?2 has the form Find a and b such that is rotated into

Problem 22E Let A be a 3 × 3 matrix with the property that the linear transformation x ? Ax maps ?3 onto ?3. Explain why the transformation must be one-to-one.