Suppose A is a 4 × 4 matrix and B is a 4 × 2 matrix, and

Problem 2E Find a basis for the set of all 2 vectors of the form

Problem 22E Determine if the matrix pairs in Exercises 19–22 are controllable.

Problem 3E and Find an implicit description of W ; that is, find a set of one or more homogeneous equations that characterize the points of W . [Hint: When is b in W?

Problem 4E Explain what is wrong with the following discussion: Let Then {f, g} is linearly dependent because g is a multiple of f.

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.) In parts are vectors in a nonzero finite-dimensional vector space V, and a. The set of all linear combinations of is a vector space b. spans V , then S spans V . c. is linearly independent, then so is S. d. If S is linearly independent, then S is a basis for V. e. If Span S = V , then some subset of S is a basis for V . f. If dim V = p and Span S = V , then S cannot be linearly dependent. g. A plane in is a two-dimensional subspace. h. The nonpivot columns of a matrix are always linearly dependent. i. Row operations on a matrix A can change the linear dependence relations among the rows of A. j. Row operations on a matrix can change the null space. k. The rank of a matrix equals the number of nonzero rows. l. If an m × n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of . m. If B is obtained from a matrix A by several elementary row operations, then rank B = rank A. n. The nonzero rows of a matrix A form a basis for Row A. o. If matrices A and B have the same reduced echelon form, then Row A = Row B. p. If H is a subspace of , then there is a 3 × 3 matrix A such that H = Col A. q. If A is m × n and rank A = m, then the linear transformation is one-to-one. r. If A is m × n and the linear transformation is onto, then rank A = m. s. A change-of-coordinates matrix is always invertible. t. are bases for a vector space V , then the j th column of the change-of-coordinates matrix .

Problem 6E Suppose p1, p2, p3, p4 are specific polynomials that span a two-dimensional subspace H of . Describe how one can find a basis for H by examining the four polynomials and making almost no computations.

Problem 5E Consider the polynomials Use the method described in the proof of the Spanning Set Theorem (Section 4.3) to produce a basis for H. (Explain how to select appropriate members of S.)

Problem 8E Let H be an n-dimensional subspace of an n-dimensional vector space V . Explain why H = V

Problem 7E What would you have to know about the solution set of a homogeneous system of 18 linear equations in 20 variables in order to know that every associated nonhomogeneous equation has a solution? Discuss.

Problem 9E Let be a linear transformation. a. What is the dimension of the range of T if T is a one-to-one mapping? Explain. b. What is the dimension of the kernel of T (see Section 4.2) if T maps onto ? Explain.

Exercises 12–17 develop properties of rank that are sometimes needed in applications. Assume the matrix A is \(m \times n\). Show from parts (a) and (b) that rank AB cannot exceed the rank of A or the rank of B. (In general, the rank of a product of matrices cannot exceed the rank of any factor in the product.) a. Show that if B is \(n \times p\), then rank \(A B \leq \operatorname{rank} A\). [Hint: Explain why every vector in the column space of AB is in the column space of A.] b. Show that if B is \(n \times p\), then rank \(A B \leq \operatorname{rank} B\). [Hint: Use part (a) to study rank \((A B)^{T}\).]

Problem 11E Let S be a finite minimal spanning set of a vector space V . That is, S has the property that if a vector is removed from S, then the new set will no longer span V . Prove that S must be a basis for V.

Problem 10E Let S be a maximal linearly independent subset of a vector space V. That is, S has the property that if a vector not in S is adjoined to S, then the new set will no longer be linearly independent. Prove that S must be a basis for V . [Hint: What if S were linearly independent but not a basis of V?

Problem 14E Exercises 12–17 develop properties of rank that are sometimes needed in applications. Assume the matrix A is m × n. Show that if Q is invertible, then rank AQ = rank A. [Hint: Use Exercise 13 to study rank (AQ)T .] Exercise 13: Show that if P is an invertible m × m matrix, then rank PA = rank A. [Hint: Apply Exercise 12 to PA and P–1 (PA)] Exercise 12: Show from parts (a) and (b) that rank AB cannot exceed the rank of A or the rank of B. (In general, the rank of a product of matrices cannot exceed the rank of any factor in the product.) a. Show that if B is n × p, then rank AB rank A. [Hint: Explain why every vector in the column space of AB is in the column space of A.] b. Show that if B is n × p, then rank AB rank B. [Hint: Use part (a) to study rank.(AB)T .]

Exercises 12–17 develop properties of rank that are sometimes needed in applications. Assume the matrix A is \(m \times n\). Show that if P is an invertible \(m \times m\) matrix, then rank PA = rank A. [Hint: Apply Exercise 12 to PA and \(P^{-1}(P A)\).]

Problem 15E Exercises 12–17 develop properties of rank that are sometimes needed in applications. Assume the matrix A is m × n. Let A be an m × n matrix, and let B be an n × p matrix such that AB = 0. Show that rank A + rank B n. [Hint: One of the four subspaces Nul A, Col A, Nul B, and Col B is contained in one of the other three subspaces.]

Problem 17E Exercises 12–17 develop properties of rank that are sometimes needed in applications. Assume the matrix A is m × n. A submatrix of a matrix A is any matrix that results from deleting some (or no) rows and/or columns of A. It can be shown that A has rank r if and only if A contains an invertible r × r submatrix and no larger square submatrix is invertible. Demonstrate part of this statement by explaining (a) why an m × n matrix A of rank r has an m × r submatrix A1 of rank r, and (b) why A1 has an invertible r × r submatrix A2. The concept of rank plays an important role in the design of engineering control systems, such as the space shuttle system mentioned in this chapter’s introductory example. A state-space model of a control system includes a difference equation of the form where is a sequence of “state vectors” in that describe the state of the system at discrete times, and is a control, or input, sequence. The pair (A, B) is said to be controllable if The matrix that appears in (2) is called the controllability matrix for the system. If (A, B) is controllable, then the system can be controlled, or driven from the state 0 to any specified state v (in ) in at most n steps, simply by choosing an appropriate control sequence in . This fact is illustrated in Exercise 18 for n = 4 and m = 2. For a further discussion of controllability, see this text’s web site (Case Study for Chapter 4).

Exercises 12–17 develop properties of rank that are sometimes needed in applications. Assume the matrix A is \(m \times n\). If A is an \(m \times n\) matrix of rank r, then a rank factorization of A is an equation of the form A = CR, where C is an \(m \times r\) matrix of rank r and R is an \(r \times n\) matrix of rank r. Such a factorization always exists (Exercise 38 in Section 4.6). Given any two \(m \times n\) matrices A and B, use rank factorizations of A and B to prove that rank \((A+B) \leq \operatorname{rank} A+\operatorname{rank} B\) [Hint: Write A + B as the product of two partitioned matrices].

Problem 19E Determine if the matrix pairs in Exercises 19–22 are controllable.

Determine if the matrix pairs in Exercises 19–22 are controllable. \([\mathbf{M}] A=\left[\begin{array}{rccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -2 & -4.2 & -4.8 & -3.6 \end{array}\right]\), \(B=\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ -1 \end{array}\right]\)

Determine if the matrix pairs in Exercises 19–22 are controllable. \(A=\left[\begin{array}{rrr}.8 & -.3 & 0 \\ .2 & .5 & 1 \\ 0 & 0 & -.5\end{array}\right], B=\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right]\)

Problem 18E Suppose A is a 4 × 4 matrix and B is a 4 × 2 matrix, and let represent a sequence of input vectors in a. Set from equation (1), and write a formula for x4 involving the controllability matrix M appearing in equation (2). (Note: The matrix M is constructed as a partitioned matrix. Its overall size here is 4 × 8.) b. Suppose (A, B) is controllable and v is any vector in . Explain why there exists a control sequence in such that x4 = v.