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Chapter 1: Q36E (page 1)

In Exercises 32鈥�36, column vectors are written as rows, such as \({\bf{x}} = \left( {{x_1},{x_2}} \right)\), and \(T\left( {\bf{x}} \right)\) is written as \(T\left( {{x_1},{x_2}} \right)\).

36.Let \(T:{\mathbb{R}^3} \to {\mathbb{R}^3}\) be the transformation that projects each vector \({\bf{x}} = \left( {{x_1},{x_2},{x_3}} \right)\) onto the plane \({x_2} = 0\), so \(T\left( {\bf{x}} \right) = T\left( {{x_1},0,{x_3}} \right)\). Show that T is a linear transformation.

Short Answer

Expert verified

\(T\) is a linear transformation.

Step by step solution

01

Write the condition for the transformation to be linear

The transformation \(T\) is said to be linear if all vectors \({\bf{u}}\) in \({\mathbb{R}^n}\) and all scalars \(c\) and \(d\) are represented in the domain \(T\)as shown below:

  • \(T\left( {c{\bf{u}}} \right) = cT\left( {\bf{u}} \right)\)
  • \(T\left( {c{\bf{u}} + d{\bf{v}}} \right) = cT\left( {\bf{u}} \right) + dT\left( {\bf{v}} \right)\)
02

Obtain the linear combination of vectors \(c{\bf{u}} + d{\bf{v}}\)

Let\({\bf{u}} = \left( {{u_1},{u_2},{u_3}} \right)\), and\({\bf{v}} = \left( {{v_1},{v_2},{v_3}} \right)\).

Substitute\({\bf{u}} = \left( {{u_1},{u_2},{u_3}} \right)\), and\({\bf{v}} = \left( {{v_1},{v_2},{v_3}} \right)\)in\(c{\bf{u}} + d{\bf{v}}\) as shown below:

\(\begin{aligned}{c}c{\bf{u}} + d{\bf{v}} &= c\left( {{u_1},{u_2},{u_3}} \right) + d\left( {{v_1},{v_2},{v_3}} \right)\\ &= \left( {c{u_1},c{u_2},c{u_3}} \right) + \left( {d{v_1},d{v_2},d{v_3}} \right)\\ &= \left( {c{u_1} + d{v_1},c{u_2} + d{v_2},c{u_3} + d{v_3}} \right)\end{aligned}\)

03

Obtain the transformation \(T\left( {c{\bf{u}} + d{\bf{v}}} \right)\)

As the vector is \(c{\bf{u}} + d{\bf{v}} = \left( {c{u_1} + d{v_1},c{u_2} + d{v_2},c{u_3} + d{v_3}} \right)\); apply the transformation by using the concept that for\({\bf{x}} = \left( {{x_1},{x_2},{x_3}} \right)\), the transformation is\(T\left( {\bf{x}} \right) = T\left( {{x_1},0,{x_3}} \right)\).

\(\begin{aligned}{c}T\left( {c{\bf{u}} + d{\bf{v}}} \right) &= T\left( {c{u_1} + d{v_1},c{u_2} + d{v_2},c{u_3} + d{v_3}} \right)\\ &= \left( {c{u_1} + d{v_1},0,c{u_3} + d{v_3}} \right)\\ &= \left( {c{u_1} + d{v_1},0,c{u_3} + d{v_3}} \right)\\ &= \left( {c{u_1},0,c{u_3}} \right) + \left( {d{v_1},0,d{v_3}} \right)\end{aligned}\)

Simplify further.

\(\begin{aligned}{c}T\left( {c{\bf{u}} + d{\bf{v}}} \right) &= c\left( {{u_1},0,{u_3}} \right) + d\left( {{v_1},0,{v_3}} \right)\\ &= cT\left( {\bf{u}} \right) + dT\left( {\bf{v}} \right)\end{aligned}\)

Since \(T\left( {c{\bf{u}} + d{\bf{v}}} \right) = cT\left( {\bf{u}} \right) + dT\left( {\bf{v}} \right)\), \(T\) is a linear transformation.

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Most popular questions from this chapter

Suppose Ais an \(n \times n\) matrix with the property that the equation \(A{\mathop{\rm x}\nolimits} = 0\) has at least one solution for each b in \({\mathbb{R}^n}\). Without using Theorem 5 or 8, explain why each equation Ax = b has in fact exactly one solution.

Find an equation involving \(g,\,h,\)and \(k\) that makes this augmented matrix correspond to a consistent system:

\(\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&g\\0&3&{ - 5}&h\\{ - 2}&5&{ - 9}&k\end{array}} \right]\)

In Exercise 23 and 24, make each statement True or False. Justify each answer.

24.

a. Any list of five real numbers is a vector in \({\mathbb{R}^5}\).

b. The vector \({\mathop{\rm u}\nolimits} \) results when a vector \({\mathop{\rm u}\nolimits} - v\) is added to the vector \({\mathop{\rm v}\nolimits} \).

c. The weights \({{\mathop{\rm c}\nolimits} _1},...,{c_p}\) in a linear combination \({c_1}{v_1} + \cdot \cdot \cdot + {c_p}{v_p}\) cannot all be zero.

d. When are \({\mathop{\rm u}\nolimits} \) nonzero vectors, Span \(\left\{ {u,v} \right\}\) contains the line through \({\mathop{\rm u}\nolimits} \) and the origin.

e. Asking whether the linear system corresponding to an augmented matrix \(\left[ {\begin{array}{*{20}{c}}{{{\rm{a}}_{\rm{1}}}}&{{{\rm{a}}_{\rm{2}}}}&{{{\rm{a}}_{\rm{3}}}}&{\rm{b}}\end{array}} \right]\) has a solution amounts to asking whether \({\mathop{\rm b}\nolimits} \) is in Span\(\left\{ {{a_1},{a_2},{a_3}} \right\}\).

Let \(u = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\) and \(v = \left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right]\). Show that \(\left[ {\begin{array}{*{20}{c}}h\\k\end{array}} \right]\) is in Span \(\left\{ {u,v} \right\}\) for all \(h\) and\(k\).

Determine whether the statements that follow are true or false, and justify your answer.

18: [111315171921][-13-1]=[131921]
See all solutions

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