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Chapter 1: Q34Q (page 1)

Why is the question “Is the linear transformation T onto?” an existence question?

Short Answer

Expert verified

The given question is an existence question.

Step by step solution

01

Identify the condition for onto

If the range of transformation T includes the entire codomain \({\mathbb{R}^m}\), then T is onto \({\mathbb{R}^m}\). The transformation maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\) if at least one solution exists for \(T\left( {\bf{x}} \right) = {\bf{b}}\). Also, each vector b should be in the codomain \({\mathbb{R}^m}\).

02

Complete the statement to make it true

Consider vector u is in\({\mathbb{R}^m}\). All the vectors of u must be in the codomain\({\mathbb{R}^m}\), such that\(T\left( {\bf{x}} \right) = {\bf{u}}\). Also, in\({\mathbb{R}^n}\), vector x exists such that\(T\left( {\bf{x}} \right) = {\bf{u}}\), where\(T\left( {\bf{x}} \right) = {\bf{u}}\)is the linear transformation.

Thus, the given question is an existence question.

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Most popular questions from this chapter

Let \({{\mathop{\rm a}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}1\\4\\{ - 2}\end{array}} \right],{{\mathop{\rm a}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 3}\\7\end{array}} \right],\) and \({\rm{b = }}\left[ {\begin{array}{*{20}{c}}4\\1\\h\end{array}} \right]\). For what values(s) of \(h\) is \({\mathop{\rm b}\nolimits} \) in the plane spanned by \({{\mathop{\rm a}\nolimits} _1}\) and \({{\mathop{\rm a}\nolimits} _2}\)?

If Ais an \(n \times n\) matrix and the equation \(A{\bf{x}} = {\bf{b}}\) has more than one solution for some b, then the transformation \({\bf{x}}| \to A{\bf{x}}\) is not one-to-one. What else can you say about this transformation? Justify your answer.a

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?

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31. \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\4&{ - 1}&3&{ - 6}\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\0&7&{ - 1}&{ - 6}\end{array}} \right]\)

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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