代数示例

S ([\begin{array}{c} a \\ b \\ c \end{array}]) = [\begin{array}{c} a + 3 b - 6 c \\ 2 a + b + c \\ a + 5 b + c \end{array}]

$S ([\begin{array}{c} a \\ b \\ c \end{array}]) = [\begin{array}{c} a + 3 b - 6 c \\ 2 a + b + c \\ a + 5 b + c \end{array}]$

解题步骤 1

该转换定义了从

ℝ^{3}

$ℝ^{3}$ 到

ℝ^{3}

$ℝ^{3}$ 的映射。若要证明该转换是线性转换，必须证明该转换满足标量乘法、加法和零向量。

ℝ^{3} \to ℝ^{3}

$ℝ^{3} \to ℝ^{3}$

解题步骤 2

首先证明变换将保留此性质。

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

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

解题步骤 3

建立两个矩阵以验证加法性质仍然存在于

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}])

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

解题步骤 4

将两个矩阵相加。

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

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

解题步骤 5

对矢量进行转换。

S (x + y) = [\begin{array}{c} x_{1} + y_{1} + 3 (x_{2} + y_{2}) - 6 (x_{3} + y_{3}) \\ 2 (x_{1} + y_{1}) + x_{2} + y_{2} + x_{3} + y_{3} \\ x_{1} + y_{1} + 5 (x_{2} + y_{2}) + x_{3} + y_{3} \end{array}]

$S (x + y) = [\begin{array}{c} x_{1} + y_{1} + 3 (x_{2} + y_{2}) - 6 (x_{3} + y_{3}) \\ 2 (x_{1} + y_{1}) + x_{2} + y_{2} + x_{3} + y_{3} \\ x_{1} + y_{1} + 5 (x_{2} + y_{2}) + x_{3} + y_{3} \end{array}]$

解题步骤 6

化简矩阵中的每一个元素。

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解题步骤 6.1

重新排列

x_{1} + y_{1} + 3 (x_{2} + y_{2}) - 6 (x_{3} + y_{3})

$x_{1} + y_{1} + 3 (x_{2} + y_{2}) - 6 (x_{3} + y_{3})$ 。

S (x + y) = [\begin{array}{c} x_{1} + 3 x_{2} - 6 x_{3} + y_{1} + 3 y_{2} - 6 y_{3} \\ 2 (x_{1} + y_{1}) + x_{2} + y_{2} + x_{3} + y_{3} \\ x_{1} + y_{1} + 5 (x_{2} + y_{2}) + x_{3} + y_{3} \end{array}]

$S (x + y) = [\begin{array}{c} x_{1} + 3 x_{2} - 6 x_{3} + y_{1} + 3 y_{2} - 6 y_{3} \\ 2 (x_{1} + y_{1}) + x_{2} + y_{2} + x_{3} + y_{3} \\ x_{1} + y_{1} + 5 (x_{2} + y_{2}) + x_{3} + y_{3} \end{array}]$

解题步骤 6.2

重新排列

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

$2 (x_{1} + y_{1}) + x_{2} + y_{2} + x_{3} + y_{3}$ 。

S (x + y) = [\begin{array}{c} x_{1} + 3 x_{2} - 6 x_{3} + y_{1} + 3 y_{2} - 6 y_{3} \\ 2 x_{1} + x_{2} + x_{3} + 2 y_{1} + y_{2} + y_{3} \\ x_{1} + y_{1} + 5 (x_{2} + y_{2}) + x_{3} + y_{3} \end{array}]

$S (x + y) = [\begin{array}{c} x_{1} + 3 x_{2} - 6 x_{3} + y_{1} + 3 y_{2} - 6 y_{3} \\ 2 x_{1} + x_{2} + x_{3} + 2 y_{1} + y_{2} + y_{3} \\ x_{1} + y_{1} + 5 (x_{2} + y_{2}) + x_{3} + y_{3} \end{array}]$

解题步骤 6.3

重新排列

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

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

S (x + y) = [\begin{array}{c} x_{1} + 3 x_{2} - 6 x_{3} + y_{1} + 3 y_{2} - 6 y_{3} \\ 2 x_{1} + x_{2} + x_{3} + 2 y_{1} + y_{2} + y_{3} \\ x_{1} + 5 x_{2} + x_{3} + y_{1} + 5 y_{2} + y_{3} \end{array}]

$S (x + y) = [\begin{array}{c} x_{1} + 3 x_{2} - 6 x_{3} + y_{1} + 3 y_{2} - 6 y_{3} \\ 2 x_{1} + x_{2} + x_{3} + 2 y_{1} + y_{2} + y_{3} \\ x_{1} + 5 x_{2} + x_{3} + y_{1} + 5 y_{2} + y_{3} \end{array}]$

S (x + y) = [\begin{array}{c} x_{1} + 3 x_{2} - 6 x_{3} + y_{1} + 3 y_{2} - 6 y_{3} \\ 2 x_{1} + x_{2} + x_{3} + 2 y_{1} + y_{2} + y_{3} \\ x_{1} + 5 x_{2} + x_{3} + y_{1} + 5 y_{2} + y_{3} \end{array}]

解题步骤 7

通过变量分组，将结果分解为两个矩阵。

S (x + y) = [\begin{array}{c} x_{1} + 3 x_{2} - 6 x_{3} \\ 2 x_{1} + x_{2} + x_{3} \\ x_{1} + 5 x_{2} + x_{3} \end{array}] + [\begin{array}{c} y_{1} + 3 y_{2} - 6 y_{3} \\ 2 y_{1} + y_{2} + y_{3} \\ y_{1} + 5 y_{2} + y_{3} \end{array}]

$S (x + y) = [\begin{array}{c} x_{1} + 3 x_{2} - 6 x_{3} \\ 2 x_{1} + x_{2} + x_{3} \\ x_{1} + 5 x_{2} + x_{3} \end{array}] + [\begin{array}{c} y_{1} + 3 y_{2} - 6 y_{3} \\ 2 y_{1} + y_{2} + y_{3} \\ y_{1} + 5 y_{2} + y_{3} \end{array}]$

解题步骤 8

变换的加法性质有效。

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

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

解题步骤 9

为保证变换为线性变换，必须支持标量乘法。

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

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

解题步骤 10

从每一元素对

p

$p$ 进行因式分解。

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解题步骤 10.1

将

p

$p$ 乘以矩阵中的每一个元素。

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

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

解题步骤 10.2

对矢量进行转换。

S (p x) = [\begin{array}{c} (p a) + 3 (p b) - 6 (p c) \\ 2 (p a + p b + p c) \\ (p a) + 5 (p b) + p c \end{array}]

$S (p x) = [\begin{array}{c} (p a) + 3 (p b) - 6 (p c) \\ 2 (p a + p b + p c) \\ (p a) + 5 (p b) + p c \end{array}]$

解题步骤 10.3

化简矩阵中的每一个元素。

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解题步骤 10.3.1

重新排列

(p a) + 3 (p b) - 6 (p c)

$(p a) + 3 (p b) - 6 (p c)$ 。

S (p x) = [\begin{array}{c} a p + 3 b p - 6 c p \\ 2 (p a + p b + p c) \\ (p a) + 5 (p b) + p c \end{array}]

$S (p x) = [\begin{array}{c} a p + 3 b p - 6 c p \\ 2 (p a + p b + p c) \\ (p a) + 5 (p b) + p c \end{array}]$

解题步骤 10.3.2

重新排列

2 (p a + p b + p c)

$2 (p a + p b + p c)$ 。

S (p x) = [\begin{array}{c} a p + 3 b p - 6 c p \\ 2 a p + 2 b p + 2 c p \\ (p a) + 5 (p b) + p c \end{array}]

$S (p x) = [\begin{array}{c} a p + 3 b p - 6 c p \\ 2 a p + 2 b p + 2 c p \\ (p a) + 5 (p b) + p c \end{array}]$

解题步骤 10.3.3

重新排列

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

$(p a) + 5 (p b) + p c$ 。

S (p x) = [\begin{array}{c} a p + 3 b p - 6 c p \\ 2 a p + 2 b p + 2 c p \\ a p + 5 b p + c p \end{array}]

$S (p x) = [\begin{array}{c} a p + 3 b p - 6 c p \\ 2 a p + 2 b p + 2 c p \\ a p + 5 b p + c p \end{array}]$

S (p x) = [\begin{array}{c} a p + 3 b p - 6 c p \\ 2 a p + 2 b p + 2 c p \\ a p + 5 b p + c p \end{array}]

$S (p x) = [\begin{array}{c} a p + 3 b p - 6 c p \\ 2 a p + 2 b p + 2 c p \\ a p + 5 b p + c p \end{array}]$

解题步骤 10.4

对矩阵的每一个元素进行因式分解。

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解题步骤 10.4.1

通过乘以

a p + 3 b p - 6 c p

$a p + 3 b p - 6 c p$ 对元素

0, 0

$0, 0$ 进行因式分解。

S (p x) = [\begin{array}{c} p (a + 3 b - 6 c) \\ 2 a p + 2 b p + 2 c p \\ a p + 5 b p + c p \end{array}]

$S (p x) = [\begin{array}{c} p (a + 3 b - 6 c) \\ 2 a p + 2 b p + 2 c p \\ a p + 5 b p + c p \end{array}]$

解题步骤 10.4.2

通过乘以

2 a p + 2 b p + 2 c p

$2 a p + 2 b p + 2 c p$ 对元素

1, 0

$1, 0$ 进行因式分解。

S (p x) = [\begin{array}{c} p (a + 3 b - 6 c) \\ p (2 a + 2 b + 2 c) \\ a p + 5 b p + c p \end{array}]

$S (p x) = [\begin{array}{c} p (a + 3 b - 6 c) \\ p (2 a + 2 b + 2 c) \\ a p + 5 b p + c p \end{array}]$

解题步骤 10.4.3

通过乘以

a p + 5 b p + c p

$a p + 5 b p + c p$ 对元素

2, 0

$2, 0$ 进行因式分解。

S (p x) = [\begin{array}{c} p (a + 3 b - 6 c) \\ p (2 a + 2 b + 2 c) \\ p (a + 5 b + c) \end{array}]

$S (p x) = [\begin{array}{c} p (a + 3 b - 6 c) \\ p (2 a + 2 b + 2 c) \\ p (a + 5 b + c) \end{array}]$

S (p x) = [\begin{array}{c} p (a + 3 b - 6 c) \\ p (2 a + 2 b + 2 c) \\ p (a + 5 b + c) \end{array}]

$S (p x) = [\begin{array}{c} p (a + 3 b - 6 c) \\ p (2 a + 2 b + 2 c) \\ p (a + 5 b + c) \end{array}]$

S (p x) = [\begin{array}{c} p (a + 3 b - 6 c) \\ p (2 a + 2 b + 2 c) \\ p (a + 5 b + c) \end{array}]

$S (p x) = [\begin{array}{c} p (a + 3 b - 6 c) \\ p (2 a + 2 b + 2 c) \\ p (a + 5 b + c) \end{array}]$

解题步骤 11

线性转换的第二性质已保留在此转换中。

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

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

解题步骤 12

为保证变换为线性变换，必须保留零向量。

S (0) = 0

$S (0) = 0$

解题步骤 13

对矢量进行转换。

S (0) = [\begin{array}{c} (0) + 3 (0) - 6 \cdot 0 \\ 2 (0) + 0 + 0 \\ (0) + 5 (0) + 0 \end{array}]

$S (0) = [\begin{array}{c} (0) + 3 (0) - 6 \cdot 0 \\ 2 (0) + 0 + 0 \\ (0) + 5 (0) + 0 \end{array}]$

解题步骤 14

化简矩阵中的每一个元素。

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解题步骤 14.1

重新排列

(0) + 3 (0) - 6 \cdot 0

$(0) + 3 (0) - 6 \cdot 0$ 。

S (0) = [\begin{array}{c} 0 \\ 2 (0) + 0 + 0 \\ (0) + 5 (0) + 0 \end{array}]

$S (0) = [\begin{array}{c} 0 \\ 2 (0) + 0 + 0 \\ (0) + 5 (0) + 0 \end{array}]$

解题步骤 14.2

重新排列

2 (0) + 0 + 0

$2 (0) + 0 + 0$ 。

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

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

解题步骤 14.3

重新排列

(0) + 5 (0) + 0

$(0) + 5 (0) + 0$ 。

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}]

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

解题步骤 15

零向量在该转换中得到保持。

S (0) = 0

$S (0) = 0$

解题步骤 16

因为不满足线性变换的所有三个性质，所以此变换不是线性变换。

线性变换

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