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Chapter 1: Q1E (page 1)

Let \(A = \left[ {\begin{array}{*{20}{c}}2&0\\0&2\end{array}} \right]\), and define \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\) by \(T\left( {\bf{x}} \right) = A{\bf{x}}\). Find the images under \(T\) of \({\bf{u}} = \left[ {\begin{array}{*{20}{c}}1\\{ - 3}\end{array}} \right]\), and \({\bf{v}} = \left[ {\begin{array}{*{20}{c}}a\\b\end{array}} \right]\).

Short Answer

Expert verified

The images under T of vectors u and v are \(T\left( {\bf{u}} \right) = \left[ {\begin{array}{*{20}{c}}2\\{ - 6}\end{array}} \right]\) and \(T\left( {\bf{v}} \right) = \left[ {\begin{array}{*{20}{c}}{2a}\\{2b}\end{array}} \right]\).

Step by step solution

01

Write the concept for computing images under the transformation of vectors

The multiplication of matrix\(A\)of the order\(m \times n\)and vector x gives a new vector defined as\(A{\bf{x}}\)or b.

This concept is defined by the transformation rule \(T\left( {\bf{x}} \right)\). The matrix transformation is denoted as \({\bf{x}}| \to A{\bf{x}}\).

02

Obtain the image of vector u under transformation T

Consider the transformation\(T\left( {\bf{x}} \right) = A{\bf{x}}\).

Substitute ufor x in the transformation \(T\left( {\bf{x}} \right) = A{\bf{x}}\)to obtain the image of vector u under transformation T.

\(T\left( {\bf{u}} \right) = A{\bf{u}}\)

Substitute matrix\(A = \left[ {\begin{array}{*{20}{c}}2&0\\0&2\end{array}} \right]\)and\({\bf{u}} = \left[ {\begin{array}{*{20}{c}}1\\{ - 3}\end{array}} \right]\)in\(T\left( {\bf{u}} \right) = A{\bf{u}}\), as shown below:

\(\begin{aligned}{c}T\left( {\bf{u}} \right) &= \left[ {\begin{array}{*{20}{c}}2&0\\0&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1\\{ - 3}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{2\left( 1 \right) + 0\left( { - 3} \right)}\\{0\left( 1 \right) + 2\left( { - 3} \right)}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}2\\{ - 6}\end{array}} \right]\end{aligned}\)

Thus, \(T\left( {\bf{u}} \right) = \left[ {\begin{array}{*{20}{c}}2\\{ - 6}\end{array}} \right]\).

03

Obtain the image of vector v under transformation T

Substitute vfor x in the transformation \(T\left( {\bf{x}} \right) = A{\bf{x}}\)to obtain the image of vector v under transformation T.

\(T\left( {\bf{v}} \right) = A{\bf{v}}\)

Substitute matrix\(A = \left[ {\begin{array}{*{20}{c}}2&0\\0&2\end{array}} \right]\)and\({\bf{v}} = \left[ {\begin{array}{*{20}{c}}a\\b\end{array}} \right]\)in\(T\left( {\bf{v}} \right) = A{\bf{v}}\), as shown below:

\(\begin{aligned}{c}T\left( {\bf{v}} \right) &= \left[ {\begin{array}{*{20}{c}}2&0\\0&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}a\\b\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{2\left( a \right) + 0\left( b \right)}\\{0\left( a \right) + 2\left( b \right)}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{2a}\\{2b}\end{array}} \right]\end{aligned}\)

Thus, \(T\left( {\bf{v}} \right) = \left[ {\begin{array}{*{20}{c}}{2a}\\{2b}\end{array}} \right]\).

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Most popular questions from this chapter

In Exercise 2, compute \(u + v\) and \(u - 2v\).

2. \(u = \left[ {\begin{array}{*{20}{c}}3\\2\end{array}} \right]\), \(v = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\).

Consider a dynamical systemwith two components. The accompanying sketch shows the initial state vectorx鈫�0and two eigen vectors蠀1鈫�鈥�鈥�and鈥�鈥�蠀2鈫�of A (with eigen values 位1鈫�and位2鈫�respectively). For the given values of位1鈫�and位2鈫�, draw a rough trajectory. Consider the future and the past of the system.

位1鈫�=0.9,位2鈫�=0.9

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 the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the coefficients \(c\) and \(d\)? Justify your answer.

27. \(\begin{array}{l}{x_1} + 3{x_2} = f\\c{x_1} + d{x_2} = g\end{array}\)

In (a) and (b), suppose the vectors are linearly independent. What can you say about the numbers \(a,....,f\) ? Justify your answers. (Hint: Use a theorem for (b).)

  1. \(\left( {\begin{aligned}{*{20}{c}}a\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\d\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\end{aligned}} \right)\)
  2. \(\left( {\begin{aligned}{*{20}{c}}a\\1\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\1\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\\1\end{aligned}} \right)\)
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