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Chapter 1: Q17E (page 1)

In Exercises 17-20, show that \(T\) is a linear transformation by finding a matrix that implements the mapping. Note that \({x_1}\), \({x_2}\),……… are not vectors but are enteries in vectors

\(T\left( {{x_1},{x_2},{x_3},{x_4}} \right) = \left( {0,{x_1} + {x_2},{x_2} + {x_3},{x_3} + {x_4}} \right)\)

Short Answer

Expert verified

\(\left[ {\begin{array}{*{20}{c}}0&0&0&0\\1&1&0&0\\0&1&1&0\\0&0&1&1\end{array}} \right]\)

Step by step solution

01

Express \(T\left( x \right)\) in the form of a matrix

Write the linear transformation \(T\left( x \right)\).

\(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}0\\{{x_1} + {x_2}}\\{{x_2} + {x_3}}\\{{x_3} + {x_4}}\end{array}} \right]\)

02

Solve the equation \(T\left( x \right) = Ax\)

\(\left[ {\begin{array}{*{20}{c}}0\\{{x_1} + {x_2}}\\{{x_2} + {x_3}}\\{{x_3} + {x_4}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\\{{x_4}}\end{array}} \right]\)

By matrix multiplication, the order of matrix \(A\) is \(4 \times 4\).

03

Compare the rows of the matrix

From the equation \(\left[ {\begin{array}{*{20}{c}}0\\{{x_1} + {x_2}}\\{{x_2} + {x_3}}\\{{x_3} + {x_4}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\\{{x_4}}\end{array}} \right]\), the first row of matrix \(A\) is \(\left[ {\begin{array}{*{20}{c}}0&0&0&0\end{array}} \right]\).

04

Compare the rows of the matrix

From the equation \(\left( {\begin{array}{*{20}{c}}0\\{{x_1} + {x_2}}\\{{x_2} + {x_3}}\\{{x_3} + {x_4}}\end{array}} \right) = \left( A \right)\left( {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\\{{x_4}}\end{array}} \right)\), the third row of matrix \(A\) is \(\left( {\begin{array}{*{20}{c}}0&1&1&0\end{array}} \right)\).

05

Compare the rows of the matrix

From the equation \(\left[ {\begin{array}{*{20}{c}}0\\{{x_1} + {x_2}}\\{{x_2} + {x_3}}\\{{x_3} + {x_4}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\\{{x_4}}\end{array}} \right]\), the third row of matrix \(A\) is \(\left[ {\begin{array}{*{20}{c}}0&1&1&0\end{array}} \right]\).

06

Compare the rows of the matrix

From the equation \(\left[ {\begin{array}{*{20}{c}}0\\{{x_1} + {x_2}}\\{{x_2} + {x_3}}\\{{x_3} + {x_4}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\\{{x_4}}\end{array}} \right]\), the third row of matrix \(A\) is \(\left[ {\begin{array}{*{20}{c}}0&0&1&1\end{array}} \right]\).

So, the matrix given in the equation is \(\left[ {\begin{array}{*{20}{c}}0&0&0&0\\1&1&0&0\\0&1&1&0\\0&0&1&1\end{array}} \right]\).

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Most popular questions from this chapter

Let \(A\) be a \(3 \times 3\) matrix with the property that the linear transformation \({\bf{x}} \mapsto A{\bf{x}}\) maps \({\mathbb{R}^3}\) into \({\mathbb{R}^3}\). Explain why transformation must be one-to-one.

Find the general solutions of the systems whose augmented matrices are given as

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

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 \(U\left( {\mathop{\rm v}\nolimits} \right)\)).

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

13. \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 4}&2\\0&3&5\\{ - 2}&8&{ - 4}\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}3\\{ - 7}\\{ - 3}\end{array}} \right]\)

If Ais an \(n \times n\) matrix and the transformation \({\bf{x}}| \to A{\bf{x}}\) is one-to-one, what else can you say about this transformation? Justify your answer.
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