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Chapter 4: Q17Q (page 191)

The vectors \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}1\\{ - 3}\end{array}} \right],{{\mathop{\rm v}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}2\\{ - 8}\end{array}} \right],{{\mathop{\rm v}\nolimits} _3} = \left[ {\begin{array}{*{20}{c}}{ - 3}\\7\end{array}} \right]\) span \({\mathbb{R}^2}\) but do not form a basis. Find two different ways to express \(\left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]\) as a linear combination of \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}\).

Short Answer

Expert verified

The two different ways to express \(\left( {\begin{array}{*{20}{c}}1\\1\end{array}} \right)\) as a linear combination are \(5{{\mathop{\rm v}\nolimits} _1} - 2{{\mathop{\rm v}\nolimits} _2}\) and \(10{{\mathop{\rm v}\nolimits} _1} - 3{{\mathop{\rm v}\nolimits} _2} + {{\mathop{\rm v}\nolimits} _3}\).

Step by step solution

01

Write the vector equation as an augmented matrix

To solve the vector equation \({{\mathop{\rm x}\nolimits} _1}\left[ {\begin{array}{*{20}{c}}1\\{ - 3}\end{array}} \right] + {{\mathop{\rm x}\nolimits} _2}\left[ {\begin{array}{*{20}{c}}2\\{ - 8}\end{array}} \right] + {{\mathop{\rm x}\nolimits} _3}\left[ {\begin{array}{*{20}{c}}{ - 3}\\7\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]\), convert it into the augmented matrix shown below:

\(\left[ {\begin{array}{*{20}{c}}1&2&{ - 3}&1\\{ - 3}&{ - 8}&7&1\end{array}} \right]\)

02

Apply the row operation

Perform an elementary row operation to produce the row-reduced echelon form of the matrix.

At row 2, multiply row 1 by 3 and add it to row 2.

\( \sim \left( {\begin{array}{*{20}{c}}1&2&{ - 3}&1\\0&{ - 2}&{ - 2}&4\end{array}} \right)\)

At row 2, multiply row 2 by \( - \frac{1}{2}\).

\( \sim \left( {\begin{array}{*{20}{c}}1&2&{ - 3}&1\\0&1&1&{ - 2}\end{array}} \right)\)

At row 1, multiply row 2 by 2 and subtract it from row 1.

\( \sim \left( {\begin{array}{*{20}{c}}1&0&{ - 5}&5\\0&1&1&{ - 2}\end{array}} \right)\)

Hence, you can consider \({{\mathop{\rm x}\nolimits} _1} = 5 + 5{{\mathop{\rm x}\nolimits} _3}\) and \({{\mathop{\rm x}\nolimits} _2} = - 2 - {{\mathop{\rm x}\nolimits} _3}\), where \({{\mathop{\rm x}\nolimits} _3}\) is any real number.

03

Determine two different ways to express \(\left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]\) as a linear combination

Let \({{\mathop{\rm x}\nolimits} _3} = 0\) and \({{\mathop{\rm x}\nolimits} _3} = 1\) give two different ways to express \(\left( {\begin{array}{*{20}{c}}1\\1\end{array}} \right)\) as a linear combination of the vectors \(5{{\mathop{\rm v}\nolimits} _1} - 2{{\mathop{\rm v}\nolimits} _2}\) and \(10{{\mathop{\rm v}\nolimits} _1} - 3{{\mathop{\rm v}\nolimits} _2} + {{\mathop{\rm v}\nolimits} _3}\).

Thus, the two ways to express \(\left( {\begin{array}{*{20}{c}}1\\1\end{array}} \right)\) as a linear combination are \(5{{\mathop{\rm v}\nolimits} _1} - 2{{\mathop{\rm v}\nolimits} _2}\) and \(10{{\mathop{\rm v}\nolimits} _1} - 3{{\mathop{\rm v}\nolimits} _2} + {{\mathop{\rm v}\nolimits} _3}\).

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Most popular questions from this chapter

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

17. 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 \(r \times r\) submatrix and no longer square submatrix is invertible. Demonstrate part of this statement by explaining (a) why an \(m \times n\) matrix A of rank \(r\) has an \(m \times r\) submatrix \({A_1}\) of rank \(r\), and (b) why \({A_1}\) has an invertible \(r \times r\) submatrix \({A_2}\).

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

\({{\mathop{\rm x}\nolimits} _{k + 1}} = A{{\mathop{\rm x}\nolimits} _k} + B{{\mathop{\rm u}\nolimits} _k}\)for \(k = 0,1,....\) (1)

Where \(A\) is \(n \times n\), \(B\) is \(n \times m\), \(\left\{ {{{\mathop{\rm x}\nolimits} _k}} \right\}\) is a sequence of “state vectors” in \({\mathbb{R}^n}\) that describe the state of the system at discrete times, and \(\left\{ {{{\mathop{\rm u}\nolimits} _k}} \right\}\) is a control, or input, sequence. The pair \(\left( {A,B} \right)\) is said to be controllable if

\({\mathop{\rm rank}\nolimits} \left( {\begin{array}{*{20}{c}}B&{AB}&{{A^2}B}& \cdots &{{A^{n - 1}}B}\end{array}} \right) = n\) (2)

The matrix that appears in (2) is called the controllability matrix for the system. If \(\left( {A,B} \right)\) is controllable, then the system can be controlled, or driven from the state 0 to any specified state \({\mathop{\rm v}\nolimits} \) (in \({\mathbb{R}^n}\)) in at most \(n\) steps, simply by choosing an appropriate control sequence in \({\mathbb{R}^m}\). This fact is illustrated in Exercise 18 for \(n = 4\) and \(m = 2\). For a further discussion of controllability, see this text’s website (Case study for Chapter 4).

Question 18: Suppose A is a \(4 \times 4\) matrix and B is a \(4 \times 2\) matrix, and let \({{\mathop{\rm u}\nolimits} _0},...,{{\mathop{\rm u}\nolimits} _3}\) represent a sequence of input vectors in \({\mathbb{R}^2}\).

  1. Set \({{\mathop{\rm x}\nolimits} _0} = 0\), compute \({{\mathop{\rm x}\nolimits} _1},...,{{\mathop{\rm x}\nolimits} _4}\) from equation (1), and write a formula for \({{\mathop{\rm x}\nolimits} _4}\) involving the controllability matrix \(M\) appearing in equation (2). (Note: The matrix \(M\) is constructed as a partitioned matrix. Its overall size here is \(4 \times 8\).)
  2. Suppose \(\left( {A,B} \right)\) is controllable and v is any vector in \({\mathbb{R}^4}\). Explain why there exists a control sequence \({{\mathop{\rm u}\nolimits} _0},...,{{\mathop{\rm u}\nolimits} _3}\) in \({\mathbb{R}^2}\) such that \({{\mathop{\rm x}\nolimits} _4} = {\mathop{\rm v}\nolimits} \).

In Exercise 17, Ais an \(m \times n\] matrix. Mark each statement True or False. Justify each answer.

17. a. The row space of A is the same as the column space of \({A^T}\].

b. If B is any echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for Row A.

c. The dimensions of the row space and the column space of A are the same, even if Ais not square.

d. The sum of the dimensions of the row space and the null space of A equals the number of rows in A.

e. On a computer, row operations can change the apparent rank of a matrix.

[M] Let \(A = \left[ {\begin{array}{*{20}{c}}7&{ - 9}&{ - 4}&5&3&{ - 3}&{ - 7}\\{ - 4}&6&7&{ - 2}&{ - 6}&{ - 5}&5\\5&{ - 7}&{ - 6}&5&{ - 6}&2&8\\{ - 3}&5&8&{ - 1}&{ - 7}&{ - 4}&8\\6&{ - 8}&{ - 5}&4&4&9&3\end{array}} \right]\).

  1. Construct matrices \(C\) and \(N\) whose columns are bases for \({\mathop{\rm Col}\nolimits} A\) and \({\mathop{\rm Nul}\nolimits} A\), respectively, and construct a matrix \(R\) whose rows form a basis for Row\(A\).
  2. Construct a matrix \(M\) whose columns form a basis for \({\mathop{\rm Nul}\nolimits} {A^T}\), form the matrices \(S = \left[ {\begin{array}{*{20}{c}}{{R^T}}&N\end{array}} \right]\) and \(T = \left[ {\begin{array}{*{20}{c}}C&M\end{array}} \right]\), and explain why \(S\) and \(T\) should be square. Verify that both \(S\) and \(T\) are invertible.

Justify the following equalities:

a.\({\rm{dim Row }}A{\rm{ + dim Nul }}A = n{\rm{ }}\)

b.\({\rm{dim Col }}A{\rm{ + dim Nul }}{A^T} = m\)
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