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Chapter 4: Q37E (page 191)

Let\({{\rm{S}}_{\rm{0}}}\)be the vector space of all sequences of the form\(\left( {{y_0},{y_1},{y_2},...} \right)\)and define linear transformations\(T\)and\({\rm{D}}\)from\({{\rm{S}}_{\rm{0}}}\)into\({{\rm{S}}_{\rm{0}}}\)by

\(T\left( {{y_0},{y_1},{y_2},...} \right) = \left( {{y_1},{y_2},{y_3}....} \right)\)

\(D\left( {{y_0},{y_1},{y_2},...} \right) = \left( {0,{y_0},{y_1},{y_2}....} \right)\)

Show that\(TD = I\)(the identity transformation on\({{\rm{S}}_{\rm{0}}}\)) and yet\(DT \ne I\).

Short Answer

Expert verified

The transformation \(TD\) results in the same sequence, whereas the transformation \(DT\) results in a different sequence.

\(TD = I\), whereas \(DT \ne I\).

Step by step solution

01

Compute the combined transformation \(TD\) on the sequence \(\left( {{y_0},{y_1},{y_2},...} \right)\)

Use the given linear transformations of T and D on the sequence to find the combined transformation.

\(\begin{aligned} \left( {TD} \right)\left( {{y_0},{y_1},{y_2},...} \right) &= T\left( {D\left( {{y_0},{y_1},{y_2}....} \right)} \right)\\ &= T\left( {0,\,{y_0},{y_1},{y_2},...} \right)\\ &= \left( {{y_0},{y_1},{y_2},...} \right)\end{aligned}\)

02

Compute the combined transformation \(DT\) on the sequence \(\left( {{y_0},{y_1},{y_2},...} \right)\)

Use the given linear transformations of T and D on the sequence again to find the combined transformation

\(\begin{aligned} \left( {DT} \right)\left( {{y_0},{y_1},{y_2},...} \right) &= D\left( {T\left( {{y_0},{y_1},{y_2}....} \right)} \right)\\ &= D\left( {{y_1},{y_2},{y_3},...} \right)\\ &= \left( {0,{y_0},{y_1},{y_2},...} \right)\end{aligned}\)

03

Draw a conclusion

Transformation \(TD\) results in the same sequence, whereas transformation \(DT\) results in a different sequence.

\(TD = I\), whereas\(DT \ne I\).

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Most popular questions from this chapter

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?

Define by \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}{{\mathop{\rm p}\nolimits} \left( 0 \right)}\\{{\mathop{\rm p}\nolimits} \left( 1 \right)}\end{array}} \right)\). For instance, if \({\mathop{\rm p}\nolimits} \left( t \right) = 3 + 5t + 7{t^2}\), then \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}3\\{15}\end{array}} \right)\).

  1. Show that \(T\) is a linear transformation. (Hint: For arbitrary polynomials p, q in \({{\mathop{\rm P}\nolimits} _2}\), compute \(T\left( {{\mathop{\rm p}\nolimits} + {\mathop{\rm q}\nolimits} } \right)\) and \(T\left( {c{\mathop{\rm p}\nolimits} } \right)\).)
  2. Find a polynomial p in \({{\mathop{\rm P}\nolimits} _2}\) that spans the kernel of \(T\), and describe the range of \(T\).

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.

If a\({\bf{6}} \times {\bf{3}}\)matrix A has a rank 3, find dim Nul A, dim Row A, and rank\({A^T}\).

Question 11: Let\(S\)be a finite minimal spanning set of a vector space\(V\). That is,\(S\)has the property that if a vector is removed from\(S\), then the new set will no longer span\(V\). Prove that\(S\)must be a basis for\(V\).
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