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Chapter 4: Q24E (page 191)

Let \(B = \left\{ {{v_1},...,{v_n}} \right\}\) be a linearly independent set in \({\mathbb{R}^{\bf{n}}}\). Explain why Bmust be a basis for \({\mathbb{R}^{\bf{n}}}\).

Short Answer

Expert verified

Set B is a basis for \({\mathbb{R}^n}\).

Step by step solution

01

State the invertible matrix theorem

Recall the invertible matrix theorem, which states that if the square matrix is invertible, then the columns are linearly independent, and the columns form a basis for \({\mathbb{R}^n}\).

02

Apply the invertible matrix theorem and the basis theorem

It is given that\(\left\{ {{v_1},...,{v_n}} \right\}\)are a set of vectors. It forms a square matrix of the order\(n \times n\), as shown below:

\(A = \left[ {\begin{array}{*{20}{c}}{{{\bf{v}}_1}}& \cdots &{{{\bf{v}}_n}}\end{array}} \right]\)

It is given that\(B = \left\{ {{{\bf{v}}_1},...,{{\bf{v}}_n}} \right\}\)is a linearly independent set in\({\mathbb{R}^n}\). By the invertible matrix theorem, the columns of B span\({\mathbb{R}^n}\).

According to the basis theorem, if the columns of B span\({\mathbb{R}^n}\)and the columns arelinearly independent, then B must be a basis for \({\mathbb{R}^n}\).

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Most popular questions from this chapter

Find a basis for the set of vectors in\({\mathbb{R}^{\bf{2}}}\)on the line\(y = {\bf{5}}x\).

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鈥檚 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 鈥渟tate vectors鈥� 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鈥檚 website (Case study for Chapter 4).

(calculus required) Define \(T:C\left( {0,1} \right) \to C\left( {0,1} \right)\) as follows: For f in \(C\left( {0,1} \right)\), let \(T\left( t \right)\) be the antiderivative \({\mathop{\rm F}\nolimits} \) of \({\mathop{\rm f}\nolimits} \) such that \({\mathop{\rm F}\nolimits} \left( 0 \right) = 0\). Show that \(T\) is a linear transformation, and describe the kernel of \(T\). (See the notation in Exercise 20 of Section 4.1.)

In statistical theory, a common requirement is that a matrix be of full rank. That is, the rank should be as large as possible. Explain why an m n matrix with more rows than columns has full rank if and only if its columns are linearly independent.

If the null space of an \({\bf{8}} \times {\bf{5}}\) matrix A is 2-dimensional, what is the dimension of the row space of A?
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