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Chapter 1: Q18E (page 1)

The figure shows vectors \(u\), \(v\) and \(w\) along with the images \(T\left( u \right)\) and \(T\left( v \right)\) under the action of a linear transformation \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\). Copy this figure carefully, and draw the image \(T\left( w \right)\) as accurately as possible. [Hint: First write \(w\) as a linear combination of \(u\) and \(v\).]

Short Answer

Expert verified

In the first figure, draw lines through the head of \(w\), where one is parallel to \(v\) and the other parallel to \(u\).

From the figure, it can be estimated that

\(w = u + 2v\)

Step by step solution

01

Use the graph find the relation between \(u\), \(v\), and \(w\)

In the first figure, draw lines through the head of \(w\), where one is parallel to \(v\) and the other parallel to \(u\).

From the figure, it can be estimated that

\(w = u + 2v\)

02

Find the linear transformation

As \(T\) is linear, the transformation is:

\(\begin{aligned}T\left( w \right) &= T\left( u \right) + T\left( {2v} \right)\\ &= T\left( u \right) + 2T\left( v \right)\end{aligned}\)

03

Drawing the image for \(T\left( w \right)\)

Draw \(2T\left( v \right)\) in the right figure and a line parallel to \(T\left( u \right)\) through the head of \(2T\left( v \right)\) and another line parallel to \(2T\left( v \right)\) through the head of \(T\left( u \right)\).

So, the diagonal of the parallelogram in the above figure is \(T\left( w \right)\).

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Most popular questions from this chapter

In Exercises 32, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

32. \(\left[ {\begin{array}{*{20}{c}}1&2&{ - 5}&0\\0&1&{ - 3}&{ - 2}\\0&{ - 3}&9&5\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&2&{ - 5}&0\\0&1&{ - 3}&{ - 2}\\0&0&0&{ - 1}\end{array}} \right]\)

In Exercises 6, write a system of equations that is equivalent to the given vector equation.

6. \({x_1}\left[ {\begin{array}{*{20}{c}}{ - 2}\\3\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}8\\5\end{array}} \right] + {x_3}\left[ {\begin{array}{*{20}{c}}1\\{ - 6}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0\\0\end{array}} \right]\)

An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let \({T_1},...,{T_4}\) denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodes—to the left, above, to the right, and below. For instance,

\({T_1} = \left( {10 + 20 + {T_2} + {T_4}} \right)/4\), or \(4{T_1} - {T_2} - {T_4} = 30\)

33. Write a system of four equations whose solution gives estimates

for the temperatures \({T_1},...,{T_4}\).

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