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Chapter 2: Q2.8-20E (page 93)

Determine which sets in Exercises 15-20 are bases for \({\mathbb{R}^{\bf{2}}}\) and \({\mathbb{R}^{\bf{3}}}\). Justify each answer.

\(\left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{6}}}\\{\bf{7}}\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}{\bf{3}}\\{ - {\bf{4}}}\\{\bf{7}}\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{\bf{7}}\\{\bf{5}}\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{8}}\\{\bf{9}}\end{array}} \right]\)

Short Answer

Expert verified

The vectors are not the basis for any subspace.

Step by step solution

01

Form a matrix using the given column vectors

\(A = \left[ {\begin{array}{*{20}{c}}1&3&{ - 2}&0\\{ - 6}&{ - 4}&7&8\\7&7&5&9\end{array}} \right]\)

02

Analyze the matrix formed using the vectors

When a set contains more vectors than there are entries in each vector, it is linearly dependent.

So, the vectors are not the basis for any subspace.

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Most popular questions from this chapter

In Exercises 1鈥�9, assume that the matrices are partitioned conformably for block multiplication. Compute the products shown in Exercises 1鈥�4.

2. \[\left[ {\begin{array}{*{20}{c}}E&{\bf{0}}\\{\bf{0}}&F\end{array}} \right]\left[ {\begin{array}{*{20}{c}}A&B\\C&D\end{array}} \right]\]

Describe in words what happens when you compute \({A^{\bf{5}}}\), \({A^{{\bf{10}}}}\), \({A^{{\bf{20}}}}\), and \({A^{{\bf{30}}}}\) for \(A = \left( {\begin{aligned}{*{20}{c}}{1/6}&{1/2}&{1/3}\\{1/2}&{1/4}&{1/4}\\{1/3}&{1/4}&{5/12}\end{aligned}} \right)\).

Use matrix algebra to show that if A is invertible and D satisfies \(AD = I\) then \(D = {A^{ - {\bf{1}}}}\).

2. Find the inverse of the matrix \(\left( {\begin{aligned}{*{20}{c}}{\bf{3}}&{\bf{2}}\\{\bf{7}}&{\bf{4}}\end{aligned}} \right)\).

Let \(A = \left( {\begin{aligned}{*{20}{c}}3&{ - 6}\\{ - 1}&2\end{aligned}} \right)\). Construct a \({\bf{2}} \times {\bf{2}}\) matrix Bsuch that ABis the zero matrix. Use two different nonzero columns for B.
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