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Chapter 1: Q38Q (page 1)

Suppose a linear transformation \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) has the property that \(T\left( {\mathop{\rm u}\nolimits} \right) = T\left( {\mathop{\rm v}\nolimits} \right)\) for some pair of distinct vectors u and v in \({\mathbb{R}^n}\). Can Tmap \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\)? Why or why not?

Short Answer

Expert verified

The linear transformation T cannot map \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\).

Step by step solution

01

State the standard matrix of T

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation, and Abe the standard matrix for T. Then, according to theorem 12,

  1. Tmaps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\) if and only if the columns of Aspan \({\mathbb{R}^m}\);
  2. T is one-to-one if and only if the columns of Aare linearly independent.

Consider Ais the standard matrix of T.

T is not one-to-one mapping according to the hypothesis. Therefore, according to theorem 12, the standard matrix A of Thas linearly dependent columns.

02

Determine whether T   maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\)

The columns of A do not span \({\mathbb{R}^n}\) because Ais a square matrix. Therefore, the linear transformation T cannot map \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\).

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Most popular questions from this chapter

In Exercises 5, write a system of equations that is equivalent to the given vector equation.

5. \({x_1}\left[ {\begin{array}{*{20}{c}}6\\{ - 1}\\5\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}{ - 3}\\4\\0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 7}\\{ - 5}\end{array}} \right]\)

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

the given system of equations.

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

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

Use Theorem 7 in section 1.7 to explain why the columns of the matrix Aare linearly independent.

\(A = \left( {\begin{aligned}{*{20}{c}}1&0&0&0\\2&5&0&0\\3&6&8&0\\4&7&9&{10}\end{aligned}} \right)\)

In Exercises 13 and 14, determine if \({\mathop{\rm b}\nolimits} \) is a linear combination of the vectors formed from the columns of the matrix \(A\).

14. \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 2}&{ - 6}\\0&3&7\\1&{ - 2}&5\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}{11}\\{ - 5}\\9\end{array}} \right]\)

Let T be a linear transformation that maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\). Is \({T^{ - 1}}\) also one-to-one?

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