线性代数示例

$T [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} x \\ y \\ 8 x + 4 y \end{array}]$

解题步骤 1

变换核是使变换等于零向量（变换厡像）的一个向量。

$[\begin{array}{c} x \\ y \\ 8 x + 4 y \end{array}] = 0$

解题步骤 2

从向量方程创建一个方程组。

$x = 0$

$y = 0$

$8 x + 4 y = 0$

解题步骤 3

Write the system as a matrix.

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 8 & 4 & 0 \end{array}]$

解题步骤 4

求行简化阶梯形矩阵。

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解题步骤 4.1

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

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解题步骤 4.1.1

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 8 - 8 \cdot 1 & 4 - 8 \cdot 0 & 0 - 8 \cdot 0 \end{array}]$

解题步骤 4.1.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 4 & 0 \end{array}]$

解题步骤 4.2

Perform the row operation $R_{3} = R_{3} - 4 R_{2}$ to make the entry at $3, 2$ a $0$ .

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解题步骤 4.2.1

Perform the row operation $R_{3} = R_{3} - 4 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 - 4 \cdot 0 & 4 - 4 \cdot 1 & 0 - 4 \cdot 0 \end{array}]$

解题步骤 4.2.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}]$

解题步骤 5

Use the result matrix to declare the final solution to the system of equations.

$x = 0$

$y = 0$

$0 = 0$

解题步骤 6

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 0 \\ 0 \end{array}]$

解题步骤 7

Write as a solution set.

${[\begin{array}{c} 0 \\ 0 \end{array}]}$

解题步骤 8

$T$ 的核心是子空间 ${[\begin{array}{c} 0 \\ 0 \end{array}]}$ 。

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

线性代数 示例

线性代数示例