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Chapter 4: Q23E (page 191)

Suppose \({\mathbb{R}^4} = Span\left\{ {{v_1},...,{v_4}} \right\}\). Explain why \(\left\{ {{v_1},...,{v_4}} \right\}\) is a basis for \({\mathbb{R}^4}\).

Short Answer

Expert verified

The set \(\left\{ {{v_1},...,{v_4}} \right\}\) is a basis for \({\mathbb{R}^4}\).

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

State the basis theorem

It is given that\(\left\{ {{v_1},...,{v_4}} \right\}\)is the set of vectors. It forms a square matrix of the order\(4 \times 4\)as shown below:

\(A = \left[ {\begin{array}{*{20}{c}}{{{\bf{v}}_1}}&{{{\bf{v}}_2}}&{{{\bf{v}}_3}}&{{{\bf{v}}_4}}\end{array}} \right]\)

As\({\mathbb{R}^4} = {\rm{Span}}\left\{ {{v_1},...,{v_4}} \right\}\), the columns of A span\({\mathbb{R}^4}\). Also, the columns are linearly independent and form a basis.

Thus, \(\left\{ {{v_1},...,{v_4}} \right\}\) is a basis for \({\mathbb{R}^4}\).

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Most popular questions from this chapter

In Exercise 18, Ais an \(m \times n\) matrix. Mark each statement True or False. Justify each answer.

18. a. If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.

b. Row operations preserve the linear dependence relations among the rows of A.

c. The dimension of the null space of A is the number of columns of A that are not pivot columns.

d. The row space of \({A^T}\) is the same as the column space of A.

e. If A and B are row equivalent, then their row spaces are the same.

Find a basis for the set of vectors in\({\mathbb{R}^{\bf{3}}}\)in the plane\(x + {\bf{2}}y + z = {\bf{0}}\). (Hint:Think of the equation as a “system” of homogeneous equations.)

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.

Consider the polynomials , and \({p_{\bf{3}}}\left( t \right) = {\bf{2}}\) \({p_{\bf{1}}}\left( t \right) = {\bf{1}} + t,{p_{\bf{2}}}\left( t \right) = {\bf{1}} - t\)(for all t). By inspection, write a linear dependence relation among \({p_{\bf{1}}},{p_{\bf{2}}},\) and \({p_{\bf{3}}}\). Then find a basis for Span\(\left\{ {{p_{\bf{1}}},{p_{\bf{2}}},{p_{\bf{3}}}} \right\}\).

In Exercise 1, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

1. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{3}}\\{ - {\bf{5}}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - {\bf{4}}}\\{\bf{6}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{3}}\end{array}} \right)\)
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