To sketch: (i) The properties, (ii) Linear transformation. Let T:R2→R2 be the linear transformation that reflects each point through the x1axis. Let A=[100−1]

Assume the value of u=[24],v=[63]. Consider some random points for u and v. The value of u + v. u+v=[24]+[63] =[87] The transformation x↦Ax reflects points through the x1 -axis. Determine the value of T(u): T(u)=Au =[100−1][24] =[2−4] Determine the value of T(v): T(v)=Av =[100−1][63] =[6−3] Determine the value of T(u + v): T(u + v)=A(u + v) =[100−1][87] =[8−7] Show the values of the vectors as in Figure 1. Linear transformation: Consider the value of c as 2. Thus the value of cu is: cu=2×[24]=[48] Determine the transformation of cu. T(cu)=[100−1][48] =[4−8] Show the linear transformation of T as in Figure 2.

"To sketch: (i) The properties, (ii) Linear transformation...." was answered by our experts

preprekomW

Answered question

2021-02-03

To sketch:
(i) The properties,
(ii) Linear transformation.
Let

T : R^{2} \to R^{2}

be the linear transformation that reflects each point through the

x_{1} a x i s .

Let

A = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]

Answer & Explanation

wheezym

Skilled2021-02-04Added 103 answers

Assume the value of $u = [\begin{matrix} 2 \\ 4 \end{matrix}], v = [\begin{matrix} 6 \\ 3 \end{matrix}] .$
Consider some random points for u and v.
The value of $u + v .$
$u + v = [\begin{matrix} 2 \\ 4 \end{matrix}] + [\begin{matrix} 6 \\ 3 \end{matrix}]$
$= [\begin{matrix} 8 \\ 7 \end{matrix}]$
The transformation $x \mapsto A x reflects points through the x_{1}$ -axis.
Determine the value of $T (u) :$
$T (u) = A u$
$= [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] [\begin{matrix} 2 \\ 4 \end{matrix}]$
$= [\begin{matrix} 2 \\ - 4 \end{matrix}]$
Determine the value of $T (v) :$
$T (v) = A v$
$= [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] [\begin{matrix} 6 \\ 3 \end{matrix}]$
$= [\begin{matrix} 6 \\ - 3 \end{matrix}]$
Determine the value of $T (u + v) :$
$T (u + v) = A (u + v)$
$= [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] [\begin{matrix} 8 \\ 7 \end{matrix}]$
$= [\begin{matrix} 8 \\ - 7 \end{matrix}]$
Show the values of the vectors as in Figure 1.

Linear transformation:
Consider the value of c as 2.
Thus the value of cu is:
$c u = 2 \times [\begin{matrix} 2 \\ 4 \end{matrix}] = [\begin{matrix} 4 \\ 8 \end{matrix}]$
Determine the transformation of cu.
$T (c u) = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] [\begin{matrix} 4 \\ 8 \end{matrix}]$
$= [\begin{matrix} 4 \\ - 8 \end{matrix}]$
Show the linear transformation of T as in Figure 2.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Integral Calculus

Didn't find what you were looking for?