Contoh

S ([\begin{array}{c} a \\ b \\ c \end{array}]) = [\begin{array}{c} a - b - c \\ a - b + c \\ a + b + 5 c \end{array}]

$S ([\begin{array}{c} a \\ b \\ c \end{array}]) = [\begin{array}{c} a - b - c \\ a - b + c \\ a + b + 5 c \end{array}]$

Langkah 1

Transformasi mendefinisikan pemetaan dari

ℝ^{3}

$ℝ^{3}$ ke

ℝ^{3}

$ℝ^{3}$ . Untuk membuktikan transformasinya linear, transformasinya harus mempertahankan perkalian skalar, penjumlahan, dan vektor nol.

ℝ^{3} \to ℝ^{3}

$ℝ^{3} \to ℝ^{3}$

Langkah 2

Pertama, buktikan transformasi yang mempertahankan sifat ini.

S (x + y) = S (x) + S (y)

$S (x + y) = S (x) + S (y)$

Langkah 3

Buat dua matriks untuk menguji sifat penjumlahan dipertahankan untuk

S

$S$ .

S ([\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}] + [\begin{array}{c} y_{1} \\ y_{2} \\ y_{3} \end{array}])

$S ([\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}] + [\begin{array}{c} y_{1} \\ y_{2} \\ y_{3} \end{array}])$

Langkah 4

Jumlahkan kedua matriks tersebut.

S [\begin{array}{c} x_{1} + y_{1} \\ x_{2} + y_{2} \\ x_{3} + y_{3} \end{array}]

$S [\begin{array}{c} x_{1} + y_{1} \\ x_{2} + y_{2} \\ x_{3} + y_{3} \end{array}]$

Langkah 5

Terapkan transformasi ke vektor.

S (x + y) = [\begin{array}{c} x_{1} + y_{1} - (x_{2} + y_{2}) - (x_{3} + y_{3}) \\ x_{1} + y_{1} - (x_{2} + y_{2}) + x_{3} + y_{3} \\ x_{1} + y_{1} + x_{2} + y_{2} + 5 (x_{3} + y_{3}) \end{array}]

$S (x + y) = [\begin{array}{c} x_{1} + y_{1} - (x_{2} + y_{2}) - (x_{3} + y_{3}) \\ x_{1} + y_{1} - (x_{2} + y_{2}) + x_{3} + y_{3} \\ x_{1} + y_{1} + x_{2} + y_{2} + 5 (x_{3} + y_{3}) \end{array}]$

Langkah 6

Sederhanakan setiap elemen dalam matriks.

Ketuk untuk lebih banyak langkah...

Langkah 6.1

Susun kembali

x_{1} + y_{1} - (x_{2} + y_{2}) - (x_{3} + y_{3})

$x_{1} + y_{1} - (x_{2} + y_{2}) - (x_{3} + y_{3})$ .

S (x + y) = [\begin{array}{c} x_{1} - x_{2} - x_{3} + y_{1} - y_{2} - y_{3} \\ x_{1} + y_{1} - (x_{2} + y_{2}) + x_{3} + y_{3} \\ x_{1} + y_{1} + x_{2} + y_{2} + 5 (x_{3} + y_{3}) \end{array}]

$S (x + y) = [\begin{array}{c} x_{1} - x_{2} - x_{3} + y_{1} - y_{2} - y_{3} \\ x_{1} + y_{1} - (x_{2} + y_{2}) + x_{3} + y_{3} \\ x_{1} + y_{1} + x_{2} + y_{2} + 5 (x_{3} + y_{3}) \end{array}]$

Langkah 6.2

Susun kembali

x_{1} + y_{1} - (x_{2} + y_{2}) + x_{3} + y_{3}

$x_{1} + y_{1} - (x_{2} + y_{2}) + x_{3} + y_{3}$ .

S (x + y) = [\begin{array}{c} x_{1} - x_{2} - x_{3} + y_{1} - y_{2} - y_{3} \\ x_{1} - x_{2} + x_{3} + y_{1} - y_{2} + y_{3} \\ x_{1} + y_{1} + x_{2} + y_{2} + 5 (x_{3} + y_{3}) \end{array}]

$S (x + y) = [\begin{array}{c} x_{1} - x_{2} - x_{3} + y_{1} - y_{2} - y_{3} \\ x_{1} - x_{2} + x_{3} + y_{1} - y_{2} + y_{3} \\ x_{1} + y_{1} + x_{2} + y_{2} + 5 (x_{3} + y_{3}) \end{array}]$

Langkah 6.3

Susun kembali

x_{1} + y_{1} + x_{2} + y_{2} + 5 (x_{3} + y_{3})

$x_{1} + y_{1} + x_{2} + y_{2} + 5 (x_{3} + y_{3})$ .

S (x + y) = [\begin{array}{c} x_{1} - x_{2} - x_{3} + y_{1} - y_{2} - y_{3} \\ x_{1} - x_{2} + x_{3} + y_{1} - y_{2} + y_{3} \\ x_{1} + x_{2} + 5 x_{3} + y_{1} + y_{2} + 5 y_{3} \end{array}]

$S (x + y) = [\begin{array}{c} x_{1} - x_{2} - x_{3} + y_{1} - y_{2} - y_{3} \\ x_{1} - x_{2} + x_{3} + y_{1} - y_{2} + y_{3} \\ x_{1} + x_{2} + 5 x_{3} + y_{1} + y_{2} + 5 y_{3} \end{array}]$

S (x + y) = [\begin{array}{c} x_{1} - x_{2} - x_{3} + y_{1} - y_{2} - y_{3} \\ x_{1} - x_{2} + x_{3} + y_{1} - y_{2} + y_{3} \\ x_{1} + x_{2} + 5 x_{3} + y_{1} + y_{2} + 5 y_{3} \end{array}]

$S (x + y) = [\begin{array}{c} x_{1} - x_{2} - x_{3} + y_{1} - y_{2} - y_{3} \\ x_{1} - x_{2} + x_{3} + y_{1} - y_{2} + y_{3} \\ x_{1} + x_{2} + 5 x_{3} + y_{1} + y_{2} + 5 y_{3} \end{array}]$

Langkah 7

Pisahkan hasilnya menjadi dua matriks dengan mengelompokkan variabel.

S (x + y) = [\begin{array}{c} x_{1} - x_{2} - x_{3} \\ x_{1} - x_{2} + x_{3} \\ x_{1} + x_{2} + 5 x_{3} \end{array}] + [\begin{array}{c} y_{1} - y_{2} - y_{3} \\ y_{1} - y_{2} + y_{3} \\ y_{1} + y_{2} + 5 y_{3} \end{array}]

$S (x + y) = [\begin{array}{c} x_{1} - x_{2} - x_{3} \\ x_{1} - x_{2} + x_{3} \\ x_{1} + x_{2} + 5 x_{3} \end{array}] + [\begin{array}{c} y_{1} - y_{2} - y_{3} \\ y_{1} - y_{2} + y_{3} \\ y_{1} + y_{2} + 5 y_{3} \end{array}]$

Langkah 8

Sifat penambahan transformasi tetap benar.

S (x + y) = S (x) + S (y)

$S (x + y) = S (x) + S (y)$

Langkah 9

Untuk transformasi menjadi linear, harus mempertahankan perkalian skalar.

S (p x) = T (p [\begin{array}{c} a \\ b \\ c \end{array}])

$S (p x) = T (p [\begin{array}{c} a \\ b \\ c \end{array}])$

Langkah 10

Faktorkan

p

$p$ dari setiap elemen.

Ketuk untuk lebih banyak langkah...

Langkah 10.1

Kalikan

p

$p$ dengan setiap elemen di dalam matriks tersebut.

S (p x) = S ([\begin{array}{c} p a \\ p b \\ p c \end{array}])

$S (p x) = S ([\begin{array}{c} p a \\ p b \\ p c \end{array}])$

Langkah 10.2

Terapkan transformasi ke vektor.

S (p x) = [\begin{array}{c} (p a) - (p b) - (p c) \\ (p a) - (p b) + p c \\ p a + p b + 5 (p c) \end{array}]

$S (p x) = [\begin{array}{c} (p a) - (p b) - (p c) \\ (p a) - (p b) + p c \\ p a + p b + 5 (p c) \end{array}]$

Langkah 10.3

Sederhanakan setiap elemen dalam matriks.

Ketuk untuk lebih banyak langkah...

Langkah 10.3.1

Susun kembali

(p a) - (p b) - (p c)

$(p a) - (p b) - (p c)$ .

S (p x) = [\begin{array}{c} a p - 1 b p - 1 c p \\ (p a) - (p b) + p c \\ p a + p b + 5 (p c) \end{array}]

$S (p x) = [\begin{array}{c} a p - 1 b p - 1 c p \\ (p a) - (p b) + p c \\ p a + p b + 5 (p c) \end{array}]$

Langkah 10.3.2

Susun kembali

(p a) - (p b) + p c

$(p a) - (p b) + p c$ .

S (p x) = [\begin{array}{c} a p - 1 b p - 1 c p \\ a p - 1 b p + c p \\ p a + p b + 5 (p c) \end{array}]

$S (p x) = [\begin{array}{c} a p - 1 b p - 1 c p \\ a p - 1 b p + c p \\ p a + p b + 5 (p c) \end{array}]$

Langkah 10.3.3

Susun kembali

p a + p b + 5 (p c)

$p a + p b + 5 (p c)$ .

S (p x) = [\begin{array}{c} a p - 1 b p - 1 c p \\ a p - 1 b p + c p \\ a p + b p + 5 c p \end{array}]

$S (p x) = [\begin{array}{c} a p - 1 b p - 1 c p \\ a p - 1 b p + c p \\ a p + b p + 5 c p \end{array}]$

S (p x) = [\begin{array}{c} a p - 1 b p - 1 c p \\ a p - 1 b p + c p \\ a p + b p + 5 c p \end{array}]

$S (p x) = [\begin{array}{c} a p - 1 b p - 1 c p \\ a p - 1 b p + c p \\ a p + b p + 5 c p \end{array}]$

Langkah 10.4

Faktorkan setiap elemen dalam matriks.

Ketuk untuk lebih banyak langkah...

Langkah 10.4.1

Elemen faktor

0, 0

$0, 0$ dengan mengalikan

a p - 1 b p - 1 c p

$a p - 1 b p - 1 c p$ .

S (p x) = [\begin{array}{c} p (a - b - c) \\ a p - 1 b p + c p \\ a p + b p + 5 c p \end{array}]

$S (p x) = [\begin{array}{c} p (a - b - c) \\ a p - 1 b p + c p \\ a p + b p + 5 c p \end{array}]$

Langkah 10.4.2

Elemen faktor

1, 0

$1, 0$ dengan mengalikan

a p - 1 b p + c p

$a p - 1 b p + c p$ .

S (p x) = [\begin{array}{c} p (a - b - c) \\ p (a - b + c) \\ a p + b p + 5 c p \end{array}]

$S (p x) = [\begin{array}{c} p (a - b - c) \\ p (a - b + c) \\ a p + b p + 5 c p \end{array}]$

Langkah 10.4.3

Elemen faktor

2, 0

$2, 0$ dengan mengalikan

a p + b p + 5 c p

$a p + b p + 5 c p$ .

S (p x) = [\begin{array}{c} p (a - b - c) \\ p (a - b + c) \\ p (a + b + 5 c) \end{array}]

$S (p x) = [\begin{array}{c} p (a - b - c) \\ p (a - b + c) \\ p (a + b + 5 c) \end{array}]$

S (p x) = [\begin{array}{c} p (a - b - c) \\ p (a - b + c) \\ p (a + b + 5 c) \end{array}]

$S (p x) = [\begin{array}{c} p (a - b - c) \\ p (a - b + c) \\ p (a + b + 5 c) \end{array}]$

S (p x) = [\begin{array}{c} p (a - b - c) \\ p (a - b + c) \\ p (a + b + 5 c) \end{array}]

$S (p x) = [\begin{array}{c} p (a - b - c) \\ p (a - b + c) \\ p (a + b + 5 c) \end{array}]$

Langkah 11

Sifat kedua dari transformasi linear dipertahankan dalam transformasi ini.

S (p [\begin{array}{c} a \\ b \\ c \end{array}]) = p S (x)

$S (p [\begin{array}{c} a \\ b \\ c \end{array}]) = p S (x)$

Langkah 12

Agar transformasi menjadi linear, vektor nol harus dipertahankan.

S (0) = 0

$S (0) = 0$

Langkah 13

Terapkan transformasi ke vektor.

S (0) = [\begin{array}{c} (0) - (0) - (0) \\ (0) - (0) + 0 \\ 0 + 0 + 5 (0) \end{array}]

$S (0) = [\begin{array}{c} (0) - (0) - (0) \\ (0) - (0) + 0 \\ 0 + 0 + 5 (0) \end{array}]$

Langkah 14

Sederhanakan setiap elemen dalam matriks.

Ketuk untuk lebih banyak langkah...

Langkah 14.1

Susun kembali

(0) - (0) - (0)

$(0) - (0) - (0)$ .

S (0) = [\begin{array}{c} 0 \\ (0) - (0) + 0 \\ 0 + 0 + 5 (0) \end{array}]

$S (0) = [\begin{array}{c} 0 \\ (0) - (0) + 0 \\ 0 + 0 + 5 (0) \end{array}]$

Langkah 14.2

Susun kembali

(0) - (0) + 0

$(0) - (0) + 0$ .

S (0) = [\begin{array}{c} 0 \\ 0 \\ 0 + 0 + 5 (0) \end{array}]

$S (0) = [\begin{array}{c} 0 \\ 0 \\ 0 + 0 + 5 (0) \end{array}]$

Langkah 14.3

Susun kembali

0 + 0 + 5 (0)

$0 + 0 + 5 (0)$ .

S (0) = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]

$S (0) = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

S (0) = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]

$S (0) = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Langkah 15

Vektor nol dipertahankan oleh transformasi.

S (0) = 0

$S (0) = 0$

Langkah 16

Karena tiga sifat transformasi linear tidak terpenuhi, maka ini bukanlah transformasi linear.

Transformasi Linear

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