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In Exercises 7-10, let \(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},{{\mathop{\rm b}\nolimits} _2}} \right\}\) and \(C = \left\{ {{{\mathop{\rm c}\nolimits} _1},{{\mathop{\rm c}\nolimits} _2}} \right\}\) be bases for \({\mathbb{R}^2}\). In each exercise, find the change-of-coordinates matrix from \(B\) to \(C\) and the change-of-coordinates matrix from \(C\) to \(B\).

10. \({{\mathop{\rm b}\nolimits} _1} = \left( {\begin{array}{*{20}{c}}7\\{ - 2}\end{array}} \right),{{\mathop{\rm b}\nolimits} _2} = \left( {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right),{{\mathop{\rm c}\nolimits} _1} = \left( {\begin{array}{*{20}{c}}4\\1\end{array}} \right),{{\mathop{\rm c}\nolimits} _2} = \left( {\begin{array}{*{20}{c}}5\\2\end{array}} \right)\).

Short Answer

Expert verified

The change-of-coordinates matrix from \(B\) to \(C\) is \(\mathop P\limits_{C \leftarrow B} = \left( {\begin{array}{*{20}{c}}8&3\\{ - 5}&{ - 2}\end{array}} \right)\). The change-of-coordinates matrix from \(C\) to \(B\) is \(\mathop P\limits_{B \leftarrow C} = \left( {\begin{array}{*{20}{c}}2&3\\{ - 5}&{ - 8}\end{array}} \right)\).

Step by step solution

01

Change-of-coordinate matrix

Let \(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}} \right\}\)and \(C = \left\{ {{{\mathop{\rm c}\nolimits} _1},...,{{\mathop{\rm c}\nolimits} _n}} \right\}\) be bases of a vector space \(V\). Then according toTheorem 15,there is a unique \(n \times n\) matrix \(\mathop P\limits_{C \leftarrow B} \) such that

\({\left( {\mathop{\rm x}\nolimits} \right)_C} = \mathop P\limits_{C \leftarrow B} {\left( {\mathop{\rm x}\nolimits} \right)_B}\). …(1)

The columns of \(\mathop P\limits_{C \leftarrow B} \) are the \(C - \)coordinate vectors of the vectors in the basis \(B\). That is, \(\mathop P\limits_{C \leftarrow B} = \left( {\begin{array}{*{20}{c}}{{{\left( {{{\mathop{\rm b}\nolimits} _1}} \right)}_C}}&{{{\left( {{{\mathop{\rm b}\nolimits} _2}} \right)}_C}}& \cdots &{{{\left( {{{\mathop{\rm b}\nolimits} _n}} \right)}_C}}\end{array}} \right)\).

02

Determine the change-of-coordinates from \(B\) to \(C\)\(\frac{1}{4}\)

Write the augmented matrix as shown below:

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

At row 1, multiply row 1 by .

At row 2, subtract row 1 from row 2.

\( \sim \left( {\begin{array}{*{20}{c}}1&{1.25}&{1.75}&{0.5}\\0&{ - 3}&{12}&{ - 9}\end{array}} \right)\)

\( \sim \left( {\begin{array}{*{20}{c}}1&{1.25}&{1.75}&{0.5}\\0&{0.75}&{ - 3.75}&{ - 1.5}\end{array}} \right)\)

At row 2, multiply row 2 by \(1.333\).

\( \sim \left( {\begin{array}{*{20}{c}}1&{1.25}&{1.75}&{0.5}\\0&1&{ - 5}&{ - 2}\end{array}} \right)\)

At row 1, multiply row 2 by \(1.25\) and subtract it from row 1.

\( \sim \left( {\begin{array}{*{20}{c}}1&0&8&3\\0&1&{ - 5}&{ - 2}\end{array}} \right)\)

Therefore, \(\mathop P\limits_{C \leftarrow B} = \left( {\begin{array}{*{20}{c}}8&3\\{ - 5}&{ - 2}\end{array}} \right)\).

Thus, the change-of-coordinates matrix from \(B\) to \(C\) is \(\mathop P\limits_{C \leftarrow B} = \left( {\begin{array}{*{20}{c}}8&3\\{ - 5}&{ - 2}\end{array}} \right)\).

03

Determine the change-of-coordinates from \(C\) to \(B\)

It is known that \({\left( {\mathop P\limits_{C \leftarrow B} } \right)^{ - 1}}\) is the matrix that converts \(C - \)coordinates into \(B - \)coordinates. That is, \({\left( {\mathop P\limits_{C \leftarrow B} } \right)^{ - 1}} = \mathop P\limits_{B \leftarrow C} \).

\(\begin{aligned} \mathop P\limits_{B \leftarrow C} &= {\left( {\mathop P\limits_{C \leftarrow B} } \right)^{ - 1}}\\ &= {\left( {\begin{array}{*{20}{c}}8&3\\{ - 5}&{ - 2}\end{array}} \right)^{ - 1}}\\ &= \frac{1}{{ - 1}}\left( {\begin{array}{*{20}{c}}{ - 2}&{ - 3}\\5&8\end{array}} \right)\\ &= \left( {\begin{array}{*{20}{c}}2&3\\{ - 5}&{ - 8}\end{array}} \right)\end{aligned}\)

Thus, the change-of-coordinates matrix from \(C\) to \(B\) is \(\mathop P\limits_{B \leftarrow C} = \left( {\begin{array}{*{20}{c}}2&3\\{ - 5}&{ - 8}\end{array}} \right)\).

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Most popular questions from this chapter

In Exercise 4, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

4. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{2}}\\{\bf{0}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{3}}\\{ - {\bf{5}}}\\{\bf{2}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{7}}}\\{\bf{3}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{4}}}\\{\bf{8}}\\{ - {\bf{7}}}\end{array}} \right)\)

The null space of a \({\bf{5}} \times {\bf{6}}\) matrix A is 4-dimensional, what is the dimension of the column space of A.

Prove theorem 3 as follows: Given an \(m \times n\) matrix A, an element in \({\mathop{\rm Col}\nolimits} A\) has the form \(Ax\) for some x in \({\mathbb{R}^n}\). Let \(Ax\) and \(A{\mathop{\rm w}\nolimits} \) represent any two vectors in \({\mathop{\rm Col}\nolimits} A\).

  1. Explain why the zero vector is in \({\mathop{\rm Col}\nolimits} A\).
  2. Show that the vector \(A{\mathop{\rm x}\nolimits} + A{\mathop{\rm w}\nolimits} \) is in \({\mathop{\rm Col}\nolimits} A\).
  3. Given a scalar \(c\), show that \(c\left( {A{\mathop{\rm x}\nolimits} } \right)\) is in \({\mathop{\rm Col}\nolimits} A\).

(M) Let \(H = {\mathop{\rm Span}\nolimits} \left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2}} \right\}\) and \(K = {\mathop{\rm Span}\nolimits} \left\{ {{{\mathop{\rm v}\nolimits} _3},{{\mathop{\rm v}\nolimits} _4}} \right\}\), where

\({{\mathop{\rm v}\nolimits} _1} = \left( {\begin{array}{*{20}{c}}5\\3\\8\end{array}} \right),{{\mathop{\rm v}\nolimits} _2} = \left( {\begin{array}{*{20}{c}}1\\3\\4\end{array}} \right),{{\mathop{\rm v}\nolimits} _3} = \left( {\begin{array}{*{20}{c}}2\\{ - 1}\\5\end{array}} \right),{{\mathop{\rm v}\nolimits} _4} = \left( {\begin{array}{*{20}{c}}0\\{ - 12}\\{ - 28}\end{array}} \right)\)

Then \(H\) and \(K\) are subspaces of \({\mathbb{R}^3}\). In fact, \(H\) and \(K\) are planes in \({\mathbb{R}^3}\) through the origin, and they intersect in a line through 0. Find a nonzero vector w that generates that line. (Hint: w can be written as \({c_1}{{\mathop{\rm v}\nolimits} _1} + {c_2}{{\mathop{\rm v}\nolimits} _2}\) and also as \({c_3}{{\mathop{\rm v}\nolimits} _3} + {c_4}{{\mathop{\rm v}\nolimits} _4}\). To build w, solve the equation \({c_1}{{\mathop{\rm v}\nolimits} _1} + {c_2}{{\mathop{\rm v}\nolimits} _2} = {c_3}{{\mathop{\rm v}\nolimits} _3} + {c_4}{{\mathop{\rm v}\nolimits} _4}\) for the unknown \({c_j}'{\mathop{\rm s}\nolimits} \).)

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.

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