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Chapter 17: Problem 11

Gegeben sind die geordnete Standardbasis $$ \begin{gathered} E_{2}=\left(\left(\begin{array}{l} 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \end{array}\right)\right) \quad \operatorname{des} \mathbb{R}^{2} \\ B=\left(\left(\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right)\right) \quad \operatorname{des} \mathbb{R}^{3} \text { und } \\ C=\left(\left(\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 0 \\ 0 \end{array}\right),\left(\begin{array}{l} 1 \\ 0 \\ 0 \\ 0 \end{array}\right)\right) \quad \operatorname{des} \mathbb{R}^{4} \end{gathered} $$ Nun betrachten wir zwei lineare Abbildungen \(\varphi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}\) und \(\psi: \mathbb{R}^{3} \rightarrow \mathbb{R}^{4}\) definiert durch $$ \varphi\left(\left(\begin{array}{l} v_{1} \\ v_{2} \end{array}\right)\right)=\left(\begin{array}{c} v_{1}-v_{2} \\ 0 \\ 2 v_{1}-v_{2} \end{array}\right) \text { und } \psi\left(\left(\begin{array}{l} v_{1} \\ v_{2} \\ v_{3} \end{array}\right)\right)=\left(\begin{array}{c} v_{1}+2 v_{3} \\ v_{2}-v_{3} \\ v_{1}+v_{2} \\ 2 v_{1}+3 v_{3} \end{array}\right) $$ Bestimmen Sie die Darstellungsmatrizen \({ }_{B} \boldsymbol{M}(\varphi)_{E_{2}}, c^{M}(\psi)_{B}\) und \(_{C} M(\psi \circ \varphi)_{E_{2}}\)

Short Answer

Expert verified
The required matrices for the given linear transformations are as follows: 1. The matrix representing the linear transformation 饾湋 under the basis 饾惛鈧� and 饾惖: _{饾惖}M(饾湋)_{饾惛_{2}} = $$\begin{bmatrix} 0 & 2 \\ 2 & -3 \\ -1 & 2 \end{bmatrix}$$ 2. The matrix representing the linear transformation 饾湏 under the basis 饾惖 and 饾惗: _{饾惗}M(饾湏)_{饾惖} = $$\begin{bmatrix} 3 & 0 & -1 \\ -3 & 1 & 2 \\ 2 & 1 & -1 \\ 0 & -1 & 2 \end{bmatrix}$$ 3. The matrix representing the composition of 饾湏 and 饾湋 (饾湏鈭橉潨�) under the basis 饾惛鈧� and 饾惗: _{饾惗}M(饾湏鈭橉潨�)_{饾惛_{2}} = $$\begin{bmatrix} 5 & -1 \\ -5 & 2 \\ 3 & -3\\ 0 & 4\end{bmatrix}$$

Step by step solution

01

Find the transformed basis vectors under \(\varphi\) and \(\psi\)

First, we need to find the transformed basis vectors under the linear transformations \(\varphi\) and \(\psi\). We will start with \(\varphi\). Evaluate \(\varphi\) on \(E_2\) basis vectors: 1. \(\varphi\left(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\right) \Rightarrow \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix}\) 2. \(\varphi\left(\begin{bmatrix} 0 \\ 1 \end{bmatrix}\right) \Rightarrow \begin{bmatrix} -1 \\ 0 \\ -1 \end{bmatrix}\) Now we will evaluate \(\psi\) on \(B\) basis vectors: 1. \(\psi\left(\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\right) \Rightarrow \begin{bmatrix} 3 \\ 0 \\ 2 \\ 5 \end{bmatrix}\) 2. \(\psi\left(\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}\right) \Rightarrow \begin{bmatrix} 1 \\ 0 \\ 2 \\ 2 \end{bmatrix}\) 3. \(\psi\left(\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\right) \Rightarrow \begin{bmatrix} 1 \\ 0 \\ 1 \\ 2 \end{bmatrix}\)
02

Represent transformed vectors in terms of the basis vectors

Next, we need to represent the transformed vectors found in Step 1 in terms of the given basis vectors for the corresponding spaces. For transformation \(\varphi\), represent the transformed vectors in terms of the basis \(B\): 1. \(\begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix}\) can be written as \(0\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} + (2)\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} - (1)\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\). 2. \(\begin{bmatrix} -1 \\ 0 \\ -1 \end{bmatrix}\) can be written as \(2\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} - (3)\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} + (2)\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\). For transformation \(\psi\), represent the transformed vectors in terms of the basis \(C\): 1. \(\begin{bmatrix} 3 \\ 0 \\ 2 \\ 5 \end{bmatrix}\) can be written as \(3\begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} - (3)\begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix} + (2)\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix} - (0)\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}\). 2. \(\begin{bmatrix} 1 \\ 0 \\ 2 \\ 2 \end{bmatrix}\) can be written as \(0\begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} + (1)\begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix} + (1)\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix} - (-1)\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}\). 3. \(\begin{bmatrix} 1 \\ 0 \\ 1 \\ 2 \end{bmatrix}\) can be written as \((-1)\begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} + (2)\begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix} - (1)\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix} + (2)\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}\). Now that we have represented the transformed vectors in terms of the basis vectors, we can construct the matrices.
03

Construct the matrices

From the coefficients found in Step 2, we can construct the matrices \(_{B}M(\varphi)_{E_{2}}\) and \(_{C}M(\psi)_{B}\). The matrix \(_{B}M(\varphi)_{E_{2}}\) is: $$\begin{bmatrix} 0 & 2 \\ 2 & -3 \\ -1 & 2 \end{bmatrix}$$ The matrix \(_{C}M(\psi)_{B}\) is: $$\begin{bmatrix} 3 & 0 & -1 \\ -3 & 1 & 2 \\ 2 & 1 & -1 \\ 0 & -1 & 2 \end{bmatrix}$$ Lastly, to find the matrix \(_{C}M(\psi\circ\varphi)_{E_{2}}\), we first compute the composition \(\psi\circ\varphi\). This can be done by applying \(\varphi\) to the basis vectors of \(E_2\) and then applying \(\psi\) to the resulting vectors: 1. \(\psi(\varphi(\begin{bmatrix} 1 \\ 0 \end{bmatrix})) \Rightarrow \psi(\begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix}) \Rightarrow \begin{bmatrix} 5 \\ 0 \\ 3 \\ 8 \end{bmatrix}\) 2. \(\psi(\varphi(\begin{bmatrix} 0 \\ 1 \end{bmatrix})) \Rightarrow \psi(\begin{bmatrix} -1 \\ 0 \\ -1 \end{bmatrix}) \Rightarrow \begin{bmatrix} 1 \\ 0 \\ -1 \\ 4 \end{bmatrix}\) Next, represent these vectors in terms of basis \(C\): 1. \(\begin{bmatrix} 5 \\ 0 \\ 3 \\ 8 \end{bmatrix}\) can be written as \((5)\begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} - (5)\begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix} + (3)\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix} - (0)\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}\). 2. \(\begin{bmatrix} 1 \\ 0 \\ -1 \\ 4 \end{bmatrix}\) can be written as \((-1)\begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} + (2)\begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix} - (3)\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix} + (4)\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}\). Using the coefficients, we can construct the matrix \(_{C}M(\psi\circ\varphi)_{E_{2}}\): $$\begin{bmatrix} 5 & -1 \\ -5 & 2 \\ 3 & -3 \\ 0 & 4 \end{bmatrix}$$ Finally, we have found all the required matrices: 1. \(_{B}M(\varphi)_{E_{2}}= \begin{bmatrix} 0 & 2 \\ 2 & -3 \\ -1 & 2\end{bmatrix}\) 2. \(_{C}M(\psi)_{B}= \begin{bmatrix} 3 & 0 & -1 \\ -3 & 1 & 2 \\ 2 & 1 & -1\\ 0 & -1 & 2\end{bmatrix}\) 3. \(_{C}M(\psi\circ\varphi)_{E_{2}}= \begin{bmatrix} 5 & -1 \\ -5 & 2 \\ 3 & -3\\ 0 & 4\end{bmatrix}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standardbasis
Standardbasis simply refers to the default or most straightforward set of basis vectors for a given vector space. Let's consider \( \mathbb{R}^2 \), where the standard basis is \( \begin{bmatrix} 1 \ 0 \end{bmatrix}, \begin{bmatrix} 0 \ 1 \end{bmatrix} \). These represent the x-axis and y-axis unit vectors respectively.

In other dimensions, such as \( \mathbb{R}^3 \) or \( \mathbb{R}^4 \), the same concept applies but with more vectors, each having a single 1 and the rest 0s. These basis vectors help in representing any vector in the space as a combination of the basis vectors. For instance:
  • Any vector in \( \mathbb{R}^2 \) can be written as \( a\begin{bmatrix} 1 \ 0 \end{bmatrix} + b\begin{bmatrix} 0 \ 1 \end{bmatrix} \).
  • In \( \mathbb{R}^3 \), a vector \( \begin{bmatrix} x \ y \ z \end{bmatrix} \) is expressed using the standard basis vectors of this space.

This simplification is crucial because it turns the process of finding a linear transformation into one of finding how these basic units map across dimensions.
Darstellungsmatrix
A Darstellungsmatrix, or representation matrix, is a matrix that describes a linear transformation with respect to a given basis. When you have a linear map, such as \( \varphi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3} \), its effect can be visualized as converting vectors from one space to another.

Here's how you get such a matrix:
  • First, apply the transformation to the standard basis vectors of the domain space.
  • Next, express the resulting vectors in terms of the basis of the codomain space.
This is exactly what was done for \( \varphi \). We found out how the standard basis vectors of \( \mathbb{R}^2 \) look like in \( \mathbb{R}^3 \) under this transformation. Then, these results are organized into a matrix.

For example, if after applying \( \varphi \), we got the vectors \( \begin{bmatrix} 1 \ 0 \ 2 \end{bmatrix} \) and \( \begin{bmatrix} -1 \ 0 \ -1 \end{bmatrix} \), the matrix represents these mappings respectively using the transformation coefficients.
Komposition von Abbildungen
Komposition von Abbildungen stands for the composition of mappings. This is when you combine two functions, \( \varphi \) and \( \psi \) in this exercise, such that the output of \( \varphi \), a vector, becomes the input to \( \psi \). The combined transformation is written as \( \psi \circ \varphi \).

When dealing with linear maps:
  • A vector from the original space \( \mathbb{R}^2 \) undergoes transformation through \( \varphi \) first.
  • The result from \( \varphi \) is then transformed again by \( \psi \).
The challenge is to find how such a composition behaves overall, which is effectively captured by a single matrix that combines both transformations. This composition matrix is constructed using matrix multiplication:

First, compute the representation matrix for \( \psi \) as it affects the outcomes of \( \varphi \); second, multiply that by the matrix of \( \varphi \). This process facilitates complex transformations broken into manageable steps, which, however, yield direct and intuitive insights into the transformations and their compounded results.

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Most popular questions from this chapter

In der Physik sind aus den verschiedensten Gr眉nden 脛nderungen des Bezugssystems - das ist ein System, auf das sich die Orts- und Zeitangaben beziehen - n枚tig. Mathematisch betrachtet ist dies eine Koordinatentransformation, also eine lineare Abbildung. Bestimmen Sie die Darstellungsmatrix bez眉glich der Standardbasis \(E_{3}\) der Koordinatentransformation, bei der das neue Bezugssystem aus dem alten durch eine Drehung um den Winkel \(\alpha\) und der Drehachse \(e_{1}\) bzw. \(e_{2}\) bzw. \(e_{3}\) entsteht.

Folgt aus der linearen Abh盲ngigkeit der Zeilen einer reellen \(11 \times 11\)-Matrix \(\boldsymbol{A}\) die lineare Abh盲ngigkeit der Spalten von \(\boldsymbol{A}\) ?

Begr眉nden Sie die auf S. 622 gemachte Behauptung: Sind \(\varphi: V \rightarrow V^{\prime}\) und \(\psi: V^{\prime} \rightarrow V^{\prime \prime}\) linear, so ist auch die Hintereinanderausf眉hrung \(\psi \circ \varphi: V \rightarrow V^{\prime \prime}\) linear, und ist \(\varphi\) eine bijektive lineare Abbildung, so ist auch \(\varphi^{-1}: V^{\prime} \rightarrow V\) eine solche.

Wir betrachten die lineare Abbildung \(\varphi: \mathbb{R}^{4} \rightarrow\) \(\mathbb{R}^{4}, v \mapsto A v\) mit der Matrix $$ A=\left(\begin{array}{cccc} 3 & 1 & 1 & -1 \\ 1 & 3 & -1 & 1 \\ 1 & -1 & 3 & 1 \\ -1 & 1 & 1 & 3 \end{array}\right) $$ Gegeben sind weiter die Vektoren' $$ a=\left(\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \end{array}\right), \quad b=\left(\begin{array}{c} 1 \\ -1 \\ -1 \\ 1 \end{array}\right) \quad \text { und } c=\left(\begin{array}{l} 4 \\ 4 \\ 4 \\ 4 \end{array}\right) $$ (a) Berechnen Sie \(\varphi(\) a \()\) und begr眉nden Sie, dass \(\boldsymbol{b}\) im Kern von \(\varphi\) liegt. Ist \(\varphi\) injektiv? (b) Bestimmen Sie die Dimensionen von Kern und Bild der linearen Abbildung \(\varphi\). (c) Bestimmen Sie Basen des Kerns und des Bildes von \(\varphi\). (d) Bestimmen Sie die Menge \(L\) aller \(\boldsymbol{v} \in \mathbb{R}^{4}\) mit \(\varphi(\boldsymbol{v})=\boldsymbol{c}\).

Wir betrachten den reellen Vektorraum \(\mathbb{R}[X]_{3}\) aller Polynome 眉ber \(\mathbb{R}\) vom Grad kleiner oder gleich 3 , und es bezeichne \(\frac{\mathrm{d}}{\mathrm{d} x}: \mathbb{R}[X]_{3} \rightarrow \mathbb{R}[X]_{3}\) die Differenziation. Weiter sei \(E=\left(1, X, X^{2}, X^{3}\right)\) die Standardbasis von \(\mathbb{R}[X]_{3}\) (a) Bestimmen Sie die Darstellungsmatrix \(_{E} \boldsymbol{M}\left(\frac{\mathrm{d}}{\mathrm{d} X}\right)_{E}\). (b) Bestimmen Sie die Darstellungsmatrix \({ }_{B} \boldsymbol{M}\left(\frac{\mathrm{d}}{\mathrm{d} x}\right)_{B}\) von \(\frac{\mathrm{d}}{\mathrm{d} X}\) bez眉glich der geordneten Basis \(B=\left(X^{3}, 3 X^{2}, 6 X, 6\right)\) von \(\mathbb{R}[X]_{3-}\)
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