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Chapter 4: Q16Q (page 191)

Determine the dimensions of \({\mathop{\rm Nul}\nolimits} A\) and \({\mathop{\rm Col}\nolimits} A\) for the matrices shown in Exercise 13-18

16. \(A = \left( {\begin{array}{*{20}{c}}3&4\\{ - 6}&{10}\end{array}} \right)\)

Short Answer

Expert verified

The dimension of \({\mathop{\rm Col}\nolimits} A\) is 2, and that of \({\mathop{\rm Nul}\nolimits} A\) is 0.

Step by step solution

01

State the condition for the dimensions of \({\mathop{\rm Nul}\nolimits} A\) and \({\mathop{\rm Col}\nolimits} A\)

Thedimension of \({\mathop{\rm Nul}\nolimits} A\) is thenumber of free variables in the equation \(A{\mathop{\rm x}\nolimits} = 0\), and the dimension of \({\mathop{\rm Col}\nolimits} A\)is thenumber of pivot columnsin \(A\).

02

Apply the row operation

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

At row 1, multiply row 1 by \(\frac{1}{3}\).

\(\left( {\begin{array}{*{20}{c}}1&{1.333}\\{ - 6}&{10}\end{array}} \right)\)

At row 2, multiply row 1 by 6 and add it to row 2.

\(\left( {\begin{array}{*{20}{c}}1&{1.333}\\0&{18}\end{array}} \right)\)

At row 2, multiply row 2 by \(\frac{1}{{18}}\).

\(\left( {\begin{array}{*{20}{c}}1&{1.333}\\0&1\end{array}} \right)\)

At row 1, multiply row 2 by \(1.3333\) and subtract it from row 1.

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

03

Determine the dimensions of \({\mathop{\rm Nul}\nolimits} A\) and \({\mathop{\rm Col}\nolimits} A\)

\({\mathop{\rm Col}\nolimits} A\)

Matrix A has two pivot columns, so the dimension of \({\mathop{\rm Col}\nolimits} A\) is 2. The equation \(A{\mathop{\rm x}\nolimits} = 0\) has only a trivial solution since there are no columns without pivots.

Therefore, \({\mathop{\rm Nul}\nolimits} A = \left\{ 0 \right\}\) and the dimension of \({\mathop{\rm Nul}\nolimits} A\) is 0.

Thus, the dimension of \({\mathop{\rm Col}\nolimits} A\) is 2, and that of \({\mathop{\rm Nul}\nolimits} A\) is 0.

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Most popular questions from this chapter

In Exercises 15 and 16, mark each statement True or False. Justify each answer. Unless stated otherwise, \(B\) is a basis for a vector space \(V\).

In Exercises 15 and 16, mark each statement True or False. Justify each answer. Unless stated otherwise, \(B\) is a basis for a vector space \(V\).

16.

  1. If\(B\)is the standard basis for\({\mathbb{R}^n}\)then the\(B\)-coordinate vector of an\({\mathop{\rm x}\nolimits} \)in\({\mathbb{R}^n}\)is x itself.
  2. The correspondence\({\left( {\mathop{\rm x}\nolimits} \right)_B} \mapsto {\mathop{\rm x}\nolimits} \)is called coordinate mapping.
  3. In some cases, a plane in\({\mathbb{R}^3}\)can be isomorphic to\({\mathbb{R}^2}\).

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Question:In Exercises 15鈥�18, find a basis for the space spanned by the given vectors,\({{\bf{v}}_{\bf{1}}}, \ldots ,{{\bf{v}}_{\bf{5}}}\).

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