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Chapter 1: Q21Q (page 1)

Let \(T:{\mathbb{R}^3} \to {\mathbb{R}^3}\) be the linear transformation that reflects each vector through the plane \({x_{\bf{2}}} = 0\). That is, \(T\left( {{x_1},{x_2},{x_3}} \right) = \left( {{x_1}, - {x_2},{x_3}} \right)\). Find the standard matrix of \(T\).

Short Answer

Expert verified

\(\left( {\begin{aligned}{*{20}{c}}1&0&0\\0&{ - 1}&0\\0&0&1\end{aligned}} \right)\)

Step by step solution

01

Find the order of matrix \(A\)

Use the equation \(T = A{\bf{x}}\). As the order of matrix \(T\) is \(3 \times 1\) and the order of \({\bf{x}}\) is \(3 \times 1\), the order of \(A\) must be \(3 \times 3\).

\(\left( {\begin{aligned}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{ - {x_2}}\\{{x_3}}\end{aligned}} \right)\)

02

Compare the rows of the matrix

Compare both sides of the equation \(\left( {\begin{aligned}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{ - {x_2}}\\{{x_3}}\end{aligned}} \right)\) to get the first row of the matrix with unknown elements as \(\left( {\begin{aligned}{*{20}{c}}1&0&0\end{aligned}} \right)\).

03

Compare the rows of the matrix

Compare both sides of the equation \(\left( {\begin{aligned}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{ - {x_2}}\\{{x_3}}\end{aligned}} \right)\) to get the second row of the matrix with unknown elements as \(\left( {\begin{aligned}{*{20}{c}}0&{ - 1}&0\end{aligned}} \right)\).

04

Compare the rows of the matrix

Compare both sides of the equation \(\left( {\begin{aligned}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{ - {x_2}}\\{{x_3}}\end{aligned}} \right)\) to get the third row of the matrix with unknown elements as \(\left( {\begin{aligned}{*{20}{c}}0&0&1\end{aligned}} \right)\).

So, the unknown matrix in the equation is \(\left( {\begin{aligned}{*{20}{c}}1&0&0\\0&{ - 1}&0\\0&0&1\end{aligned}} \right)\).

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Most popular questions from this chapter

Write the vector \(\left( {\begin{array}{*{20}{c}}5\\6\end{array}} \right)\) as the sum of two vectors, one on the line \(\left\{ {\left( {x,y} \right):y = {\bf{2}}x} \right\}\) and one on the line \(\left\{ {\left( {x,y} \right):y = x/{\bf{2}}} \right\}\).

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

If Ais a 2×2matrix with eigenvalues 3 and 4 and if localid="1668109698541" u→ is a unit eigenvector of A, then the length of vector Alocalid="1668109419151" u→cannot exceed 4.

Suppose Ais an \(n \times n\) matrix with the property that the equation \(Ax = 0\)has only the trivial solution. Without using the Invertible Matrix Theorem, explain directly why the equation \(Ax = b\) must have a solution for each b in \({\mathbb{R}^n}\).

Suppose \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2}} \right\}\) is a linearly independent set in \({\mathbb{R}^n}\). Show that \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2}} \right\}\) is also linearly independent.
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