Assume that T is a linear transformation. Find the standard

Question

Assume that T is a linear transformation. Find the standard

pogonofor9z 2021-12-26 Answered

Assume that T is a linear transformation. Find the standard matrix of T.
T:

R^{2} \to R^{4}, T (e_{1}) = (3, 1, 3, 1), and T (e_{2}) = (- 5, 6, 0, 0)

, where

e_{1} = (1, 0)

and

e_{2} = (0, 1)

A =

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Expert Answer

Joseph Fair

Answered 2021-12-27 Author has 34 answers

Let

x \in F^{n}

. Here,

[\begin{matrix} x_{1} \\ ⋮ \\ x_{n} \end{matrix}]

T (x) = T (\sum_{j} e_{j} x_{j}) = \sum_{j} T (e_{j}) x_{j}

= \sum_{j} A_{j} x_{j} = A x

So, here

A = [T (e_{1}) \dots T (e_{n})]

According to question
T:

R^{2} \to R^{4}, T (e_{1}) = (3, 1, 3, 1), and T (e_{2}) = (- 5, 6, 0, 0)

So, the standart matrix of T is

[T (e_{1}) T (e_{2})]

= [\begin{array}{cc} 3 & - 5 \\ 1 & 6 \\ 3 & 0 \\ 1 & 0 \end{array}]

- Answer

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Beverly Smith

Answered 2021-12-28 Author has 42 answers

A=

= [\begin{array}{cc} 3 & - 5 \\ 1 & 6 \\ 3 & 0 \\ 1 & 0 \end{array}]

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karton

Answered 2021-12-30 Author has 368 answers

let $T = [\begin{matrix} a & b \\ c & d \\ e & f \\ g & h \end{matrix}]$
$T (e_{1}) = (3, 1, 3, 1)$
$\Rightarrow T (\begin{matrix} 1 \\ 0 \end{matrix}) = (\begin{matrix} 3 \\ 1 \\ 3 \\ 1 \end{matrix})$
$\Rightarrow [\begin{matrix} a & b \\ c & d \\ e & f \\ g & h \end{matrix}] [\begin{matrix} 1 \\ 0 \end{matrix}] = [\begin{matrix} 3 \\ 1 \\ 3 \\ 1 \end{matrix}]$
$\Rightarrow [\begin{matrix} a \\ c \\ e \\ g \end{matrix}] = [\begin{matrix} 3 \\ 1 \\ 3 \\ 1 \end{matrix}] \Rightarrow a = 3, c = 1, e = 3, g = 1$
$T (e_{2}) = (- 5, 6, 0, 0)$
$\Rightarrow T (\begin{matrix} 1 \\ 0 \end{matrix}) = (\begin{matrix} - 5 \\ 6 \\ 0 \\ 0 \end{matrix})$
$\Rightarrow [\begin{matrix} a & b \\ c & d \\ e & f \\ g & h \end{matrix}] [\begin{matrix} 1 \\ 0 \end{matrix}] = [\begin{matrix} - 5 \\ 6 \\ 0 \\ 0 \end{matrix}]$
$\Rightarrow [\begin{matrix} b \\ d \\ f \\ h \end{matrix}] == [\begin{matrix} - 5 \\ 6 \\ 0 \\ 0 \end{matrix}] \Rightarrow b = - 5, d = 6, f = h = 0$
$\Rightarrow T = [\begin{matrix} 3 & - 5 \\ 1 & 6 \\ 3 & 0 \\ 1 & 0 \end{matrix}]$

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