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Chapter 1: Q25E (page 1)

Given \({\bf{v}} \ne 0\) and \({\bf{p}}\) in\({\mathbb{R}^{\bf{n}}}\), the line through \({\bf{p}}\) in the direction of \({\bf{v}}\) has the parametric equation\({\bf{x}} = {\bf{p}} + {\bf{tv}}\). Show that a linear transformation \({\bf{T}}:{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{n}}}\) maps this line onto another line or onto a single point (a degenerate line).

Short Answer

Expert verified

If \(T\left( v \right) = 0\) then the image is a single point \(T\left( p \right)\). If \(T\left( v \right) \ne 0\) then the image is a line through \(T\left( p \right)\) in the direction \(T\left( v \right)\).

Step by step solution

01

Find the image of the given parametric equation using the properties of a linear transformation

\(\begin{aligned} T\left( x \right) &= T\left( {p + tv} \right)\\ &= T\left( p \right) + T\left( {tv} \right)\\T\left( x \right) &= T\left( p \right) + tT\left( v \right)\end{aligned}\)

02

Determine the image if \({\bf{T}}\left( {\bf{v}} \right) = {\bf{0}}\)

Whenever \(T\left( v \right) = 0\), \(T\left( x \right) = T\left( p \right) + 0 = T\left( p \right)\).

This implies \(T\) maps the line \(x = p + tv\) onto a single point \(T\left( p \right)\).

03

Determine the image if \({\bf{T}}\left( {\bf{v}} \right) \ne {\bf{0}}\)

Whenever \(T\left( v \right) \ne 0\), \(T\left( x \right) = T\left( p \right) + tT\left( v \right)\).

That is, \(T\) maps the line \(x = p + tv\) onto another line through the point \(T\left( p \right)\) in the direction \(T\left( v \right)\).

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Most popular questions from this chapter

Let \(A = \left[ {\begin{array}{*{20}{c}}1&0&{ - 4}\\0&3&{ - 2}\\{ - 2}&6&3\end{array}} \right]\) and \(b = \left[ {\begin{array}{*{20}{c}}4\\1\\{ - 4}\end{array}} \right]\). Denote the columns of \(A\) by \({{\mathop{\rm a}\nolimits} _1},{a_2},{a_3}\) and let \(W = {\mathop{\rm Span}\nolimits} \left\{ {{a_1},{a_2},{a_3}} \right\}\).

  1. Is \(b\) in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)? How many vectors are in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)?
  2. Is \(b\) in \(W\)? How many vectors are in W.
  3. Show that \({a_1}\) is in W.[Hint: Row operations are unnecessary.]

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 \(\left\{ {\left( {x,y} \right):y = x/{\bf{2}}} \right\}\).

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鈥攖o 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 \({T_1},...,{T_4}\).

Describe the possible echelon forms of the matrix A. Use the notation of Example 1 in Section 1.2.

a. A is a \({\bf{2}} \times {\bf{3}}\) matrix whose columns span \({\mathbb{R}^{\bf{2}}}\).

b. A is a \({\bf{3}} \times {\bf{3}}\) matrix whose columns span \({\mathbb{R}^{\bf{3}}}\).

Solve the systems in Exercises 11鈥�14.

12.\(\begin{aligned}{c}{x_1} - 3{x_2} + 4{x_3} = - 4\\3{x_1} - 7{x_2} + 7{x_3} = - 8\\ - 4{x_1} + 6{x_2} - {x_3} = 7\end{aligned}\)
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