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Chapter 1: Q2E (page 1)

In Exercise 1-10, assume that\(T\)is a linear transformation. Find the standard matrix of\(T\).

\(T:{\mathbb{R}^3} \to {\mathbb{R}^2}\), \(T\left( {{e_1}} \right) = \left( {1,3} \right)\) and \(T\left( {{e_2}} \right) = \left( {4, - 7} \right) - \) and \(T\left( {{e_3}} \right) = \left( { - 5,4} \right)\)where \({e_1}\), \({e_2}\), and \({e_3}\).

Short Answer

Expert verified

\(\left[ {\begin{array}{*{20}{c}}1&4&{ - 5}\\3&{ - 7}&4\end{array}} \right]\)

Step by step solution

01

Find the value of \(T\) using linear transformation

Using linear transformation,

\(\begin{aligned}{c}T &= T\left( {{x_1}{e_1} + {x_2}{e_2} + {x_3}{e_3}} \right)\\ &= {x_1}T\left( {{e_1}} \right) + {x_2}T\left( {{e_2}} \right) + {x_3}T\left( {{e_3}} \right)\\ &= \left[ {\begin{array}{*{20}{c}}{T\left( {{e_1}} \right)}&{T\left( {{e_2}} \right)}&{T\left( {{e_3}} \right)}\end{array}} \right]x\end{aligned}\)

02

Substitute values of \(T\left( {{e_1}} \right)\), \(T\left( {{e_2}} \right)\), and \(T\left( {{e_3}} \right)\)

By the equation \(T = \left[ {\begin{array}{*{20}{c}}{T\left( {{e_1}} \right)}&{T\left( {{e_2}} \right)}&{T\left( {{e_3}} \right)}\end{array}} \right]x\),

\(T = \left[ {\begin{array}{*{20}{c}}1&4&{ - 5}\\3&{ - 7}&4\end{array}} \right]x\).

03

Find the standard matrix \(T\) for linear transformation

In the equation \(T = Ax\), the matrix \(A\) is the matrix for linear transformation \(T\).

By the equation \(T = \left[ {\begin{array}{*{20}{c}}1&4&{ - 5}\\3&{ - 7}&4\end{array}} \right]x\), \(A = \left[ {\begin{array}{*{20}{c}}1&4&{ - 5}\\3&{ - 7}&4\end{array}} \right]\).

So, the linear transformation matrix is \(\left[ {\begin{array}{*{20}{c}}1&4&{ - 5}\\3&{ - 7}&4\end{array}} \right]\).

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Most popular questions from this chapter

In Exercises 9, write a vector equation that is equivalent to

the given system of equations.

9. \({x_2} + 5{x_3} = 0\)

\(\begin{array}{c}4{x_1} + 6{x_2} - {x_3} = 0\\ - {x_1} + 3{x_2} - 8{x_3} = 0\end{array}\)

Suppose Tand Ssatisfy the invertibility equations (1) and (2), where T is a linear transformation. Show directly that Sis a linear transformation. (Hint: Given u, v in \({\mathbb{R}^n}\), let \({\mathop{\rm x}\nolimits} = S\left( {\mathop{\rm u}\nolimits} \right),{\mathop{\rm y}\nolimits} = S\left( {\mathop{\rm v}\nolimits} \right)\). Then \(T\left( {\mathop{\rm x}\nolimits} \right) = {\mathop{\rm u}\nolimits} \), \(T\left( {\mathop{\rm y}\nolimits} \right) = {\mathop{\rm v}\nolimits} \). Why? Apply Sto both sides of the equation \(T\left( {\mathop{\rm x}\nolimits} \right) + T\left( {\mathop{\rm y}\nolimits} \right) = T\left( {{\mathop{\rm x}\nolimits} + y} \right)\). Also, consider \(T\left( {cx} \right) = cT\left( x \right)\).)

Question: If A is a non-zero matrix of the form,[a-bba] then the rank of A must be 2.

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be an invertible linear transformation, and let Sand U be functions from \({\mathbb{R}^n}\) into \({\mathbb{R}^n}\) such that \(S\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) and \(\)\(U\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\). Show that \(U\left( v \right) = S\left( v \right)\) for all v in \({\mathbb{R}^n}\). This will show that Thas a unique inverse, as asserted in theorem 9. (Hint: Given any v in \({\mathbb{R}^n}\), we can write \({\mathop{\rm v}\nolimits} = T\left( {\mathop{\rm x}\nolimits} \right)\) for some x. Why? Compute \(S\left( {\mathop{\rm v}\nolimits} \right)\) and \(U\left( {\mathop{\rm v}\nolimits} \right)\)).

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation, and suppose \(T\left( u \right) = {\mathop{\rm v}\nolimits} \). Show that \(T\left( { - u} \right) = - {\mathop{\rm v}\nolimits} \).

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