Section 1.7 Homework - Revision history

Grad at 06:58, 16 November 2015

2015-11-16T06:58:06Z

Grad: Created page with "'''3.''' Find the matrix representation for $D^2 + 2D+1_{P_3}: P_3 \to P_3$ with respect to the basis $1, t, t^2, t^3$ .

''Solution'' In or..."

2015-11-09T21:10:23Z

Created page with "'''3.''' Find the matrix representation for <math> D^2 + 2D+1_{P_3}: P_3 \to P_3</math> with respect to the basis <math>1, t, t^2, t^3</math>. ''Solution'' In or..."

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'''3.''' Find the matrix representation for <math>
D^2 + 2D+1_{P_3}: P_3 \to P_3</math> with respect to the basis <math>1, t, t^2, t^3</math>. 
 

''Solution''
In order to calculate the matrix representation, we evaluate the function on each of the basis elements and then write the coordinate vector for the output of the function in terms of the same basis. In particular if we let <math>L = D^2 + 2D + 1_{P_3}</math> then: 
<math>L(1) = 0 + 2\cdot 0 + 1 = 1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}</math> 
<math>L(t) = 0 + 2\cdot 1 + t = 2+t = \begin{bmatrix} 2 \\ 1 \\ 0 \\ 0 \end{bmatrix}</math> <math>\leftarrow</math> Fixed error here 
<math>L(t^2) = 2 + 2\cdot 2t + t^2 = 2 + 4t + t^2\begin{bmatrix} 2 \\ 4 \\ 1 \\ 0 \end{bmatrix}</math> 
<math>L(t^3) = 6t + 2\cdot 3t^2 + t^3 = 6t+6t^2+t^3 = \begin{bmatrix} 0 \\ 6 \\ 6 \\ 1 \end{bmatrix}</math> 
Which gives the matrix representation: <math>\begin{bmatrix} 1 & 2 & 2 & 0\\ 0 & 1 & 4 & 6 \\ 0 & 0 & 1 & 6 \\ 0 & 0 & 0 & 1 \end{bmatrix}</math>

'''6.''' Let <math>
\begin{bmatrix} a & c \\ b & d \end{bmatrix}</math> and consider the map <math>R_A: \text{Mat}_{2 \times 2} (\mathbb{F}) \to \text{Mat}_{2 \times 2} (\mathbb{F})</math> defined by <math>R_A(X) = XA</math>. Compute the matrix representation of this linear map with respect to the basis:

<math> E_{11} = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}</math>

<math> E_{21} = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}</math>

<math> E_{12} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}</math>

<math> E_{22} = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}</math>

''Solution'' As before we evaluate the function on the basis elements and represent the outputs as coordinate vectors. 
<math>R_A(E_{11}) = E_{11} A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\begin{bmatrix} a & c \\ b & d \end{bmatrix} = \begin{bmatrix} a & c \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} a \\ 0 \\ c \\ 0 \end{bmatrix}</math> 
<math>R_A(E_{21}) = E_{21} A = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} a & c \\ b & d \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ a & c \end{bmatrix} = \begin{bmatrix} 0 \\ a \\ 0 \\ c \end{bmatrix}</math> 
<math>R_A(E_{12}) = E_{12} A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\begin{bmatrix} a & c \\ b & d \end{bmatrix} = \begin{bmatrix} b & d \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} b \\ 0 \\ d \\ 0 \end{bmatrix}</math> 
<math>R_A(E_{22}) = E_{22} A = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} a & c \\ b & d \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ b & d \end{bmatrix} = \begin{bmatrix} 0 \\ b \\ 0 \\ d \end{bmatrix}</math> 
This gives the matrix representation of <math>R_A</math> as <math>\begin{bmatrix} a & 0 & b & 0 \\ 0 & a & 0 & b \\ c & 0 & d & 0 \\ 0 & c & 0 & d\end{bmatrix}</math> <math>L(t^3) = 6t + 2\cdot 3t^2 + t^3 = 6t+6t^2+t^3 = \begin{bmatrix} 0 \\ 6 \\ 6 \\ 1 \end{bmatrix}</math> 
Which gives the matrix representation: <math>\begin{bmatrix} 1 & 2 & 2 & 0\\ 0 & 1 & 4 & 6 \\ 0 & 0 & 1 & 6 \\ 0 & 0 & 0 & 1 \end{bmatrix}</math>

'''7.''' Compute a matrix representation for <math>L: \text{Mat}_{2 \times 2}(\mathbb{F}) \to \text{Mat}_{1 \times 2}(\mathbb{F})</math> defined by: 
<math>L(X) = \begin{bmatrix} 1 & -1 \end{bmatrix} X</math> using the standard bases. 
 

''Solution''
We again calculate: 
<math>L(E_{11}) = \begin{bmatrix} 1 & -1 \end{bmatrix} E_{11} = \begin{bmatrix} 1 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}</math> 
<math>L(E_{12}) = \begin{bmatrix} 1 & -1 \end{bmatrix} E_{12} = \begin{bmatrix} 1 & -1 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}</math> 
<math>L(E_{21}) = \begin{bmatrix} 1 & -1 \end{bmatrix} E_{21} = \begin{bmatrix} 1 & -1 \end{bmatrix}\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} -1 & 0 \end{bmatrix} = \begin{bmatrix} -1 \\ 0 \end{bmatrix}</math> 
<math>L(E_{22}) = \begin{bmatrix} 1 & -1 \end{bmatrix} E_{22} = \begin{bmatrix} 1 & -1 \end{bmatrix}\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & -1 \end{bmatrix} = \begin{bmatrix} 0 \\ -1 \end{bmatrix}</math> 
This gives the matrix representation: <math>\begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \end{bmatrix}</math>

@@ Line 3: / Line 3: @@
 <br />
-''Solution''
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-In order to calculate the matrix representation, we evaluate the function on each of the basis elements and then write the coordinate vector for the output of the function in terms of the same basis. In particular if we let <math>L = D^2 + 2D + 1_{P_3}</math> then:<br />
+!Solution:
 <math>L(1) = 0 + 2\cdot 0 + 1 = 1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}</math><br />
 <math>L(t) = 0 + 2\cdot 1 + t = 2+t = \begin{bmatrix} 2 \\ 1 \\ 0 \\ 0 \end{bmatrix}</math> <math>\leftarrow</math> Fixed error here<br />
@@ Line 10: / Line 12: @@
 <math>L(t^3) = 6t + 2\cdot 3t^2 + t^3 = 6t+6t^2+t^3 = \begin{bmatrix} 0 \\ 6 \\ 6 \\ 1 \end{bmatrix}</math><br />
 Which gives the matrix representation: <math>\begin{bmatrix} 1 & 2 & 2 & 0\\ 0 & 1 & 4 & 6 \\ 0 & 0 & 1 & 6 \\ 0 & 0 & 0 & 1 \end{bmatrix}</math>
 '''6.''' Let <math>
@@ Line 22: / Line 25: @@
 <math> E_{22} = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}</math>
-''Solution'' As before we evaluate the function on the basis elements and represent the outputs as coordinate vectors.<br />
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 <math>R_A(E_{11}) = E_{11} A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\begin{bmatrix} a & c \\ b & d \end{bmatrix} = \begin{bmatrix} a & c \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} a \\ 0 \\ c \\ 0 \end{bmatrix}</math><br />
 <math>R_A(E_{21}) = E_{21} A = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} a & c \\ b & d \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ a & c \end{bmatrix} = \begin{bmatrix} 0 \\ a \\ 0 \\ c \end{bmatrix}</math><br />
@@ Line 29: / Line 35: @@
 This gives the matrix representation of <math>R_A</math> as <math>\begin{bmatrix} a & 0 & b & 0 \\ 0 & a & 0 & b \\ c & 0 & d & 0 \\ 0 & c & 0 & d\end{bmatrix}</math> <math>L(t^3) = 6t + 2\cdot 3t^2 + t^3 = 6t+6t^2+t^3 = \begin{bmatrix} 0 \\ 6 \\ 6 \\ 1 \end{bmatrix}</math><br />
 Which gives the matrix representation: <math>\begin{bmatrix} 1 & 2 & 2 & 0\\ 0 & 1 & 4 & 6 \\ 0 & 0 & 1 & 6 \\ 0 & 0 & 0 & 1 \end{bmatrix}</math>
@@ Line 35: / Line 42: @@
 <br />
-''Solution''
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-We again calculate:<br />
+!Solution:
 <math>L(E_{11}) = \begin{bmatrix} 1 & -1 \end{bmatrix} E_{11} = \begin{bmatrix} 1 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}</math><br />
 <math>L(E_{12}) = \begin{bmatrix} 1 & -1 \end{bmatrix} E_{12} = \begin{bmatrix} 1 & -1 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}</math><br />
@@ Line 42: / Line 51: @@
 <math>L(E_{22}) = \begin{bmatrix} 1 & -1 \end{bmatrix} E_{22} = \begin{bmatrix} 1 & -1 \end{bmatrix}\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & -1 \end{bmatrix} = \begin{bmatrix} 0 \\ -1 \end{bmatrix}</math><br />
 This gives the matrix representation: <math>\begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \end{bmatrix}</math>

Section 1.7 Homework - Revision history

Grad at 06:58, 16 November 2015

Grad: Created page with "'''3.''' Find the matrix representation for D^2 + 2D+1_{P_3}: P_3 \to P_3 with respect to the basis 1, t, t^2, t^3. ''Solution'' In or..."

Grad: Created page with "'''3.''' Find the matrix representation for $D^2 + 2D+1_{P_3}: P_3 \to P_3$ with respect to the basis $1, t, t^2, t^3$ .

''Solution'' In or..."