Plane matrix transformations · Extension · 平面矩阵变换 · 拓展
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| transformation matrix/trænsfɔːˈmeɪʃn ˈmeɪtrɪks/ | 变换矩阵 | biàn huàn jǔ zhèn |
Where do the basis vectors go?
- A map stretches one direction and shears another. A matrix records the transformation of each basis vector.
- This lesson studies transformation matrix 变换矩阵: A matrix whose action on a column vector defines a plane transformation.
基向量去了哪里？
- 映射会拉伸一个方向并剪切另一个方向。矩阵记录了每个基向量的变换过程。
- 本课学习变换矩阵 transformation matrix：一个作用于列向量从而定义平面变换的矩阵。
Choose the mathematical structure
- Represent a plane transformation using a 2×2 matrix acting on column vectors. The columns give the images of the two basis vectors. Combine transformations in the specified order.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
选择数学结构
- 使用一个2×2矩阵作用于列向量来表示平面变换。该矩阵的各列分别给出两个基向量的像。按照指定顺序组合变换。
- 计算前请先明确允许的输入项和单位。方程应表达关系本身，而不仅仅是记录计算器按键过程。
Which description correctly defines transformation matrix? · 以下哪项描述正确定义了变换矩阵？
A matrix whose action on a column vector defines a plane transformation. · 一个作用于列向量以定义平面变换的矩阵。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For A=[[2,1],[0,3]], A(1,2)=(4,6). A maps (1,0) to (2,0) and (0,1) to (1,3). Matrix multiplication records composition; the rightmost matrix acts first.
通过验证案例进行推导
- 将结果与初始量进行比对。代入原始关系式，或在适当情况下对比图表与数值答案。
对于 A=[[2,1],[0,3]]，A(1,2)=(4,6)。矩阵 A 将 (1,0) 映射到 (2,0)，并将 (0,1) 映射到 (1,3)。矩阵乘法记录了复合变换；最右侧的矩阵最先作用。
Plane matrix transformations · 平面矩阵变换
Represent a plane transformation using a 2×2 matrix acting on column vectors · 使用 2×2 矩阵作用于列向量来表示平面变换
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
Find the x-coordinate of A(1,2) for A=[[2,1],[0,3]]. · 求A(1,2)的x坐标，其中A=[[2,1],[0,3]]。
First coordinate=2×1+1×2=4. · 第一个坐标=2×1+1×2=4。
Test a tempting shortcut
- Matrix multiplication is usually not commutative. Treat coordinates as column vectors and apply a composition in the correct order.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The order of a pair of matrix transformations never changes the image. This claim is false. Explain which definition or assumption it violates.
检验一个诱人的捷径
- 矩阵乘法通常不满足交换律。将坐标视为列向量，并按正确顺序应用复合变换。
- 当捷径失效时，找出其违背的假设。保留精确值直到题目要求的最终舍入步骤。
一对矩阵变换的顺序永远不会改变图像。此说法是错误的。请说明它违反了哪个定义或假设。
Find its y-coordinate. · 求其y坐标。
Second coordinate=0×1+3×2=6. · 第二个坐标=0×1+3×2=6。
The order of a pair of matrix transformations never changes the image. · 两次矩阵变换的顺序不会改变像的位置。
Matrix multiplication is usually not commutative. Treat coordinates as column vectors and apply a composition in the correct order. · 矩阵乘法通常不满足交换律。将坐标视为列向量，并按正确顺序应用复合变换。
Interpret a new situation
- This 9260 Extension lesson concerns plane transformation matrices. Eigenvalues, diagonalisation and characteristic equations are outside its scope.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
解读新情境
- 本9260拓展课程涉及平面变换矩阵。特征值、对角化和特征方程不在其范围内。
- 完整解答需给出数学结果并解释其含义。检查其在所述背景下是否可行。
Find the y-coordinate of A(0,1). · 求A(0,1)的y坐标。
For input (0,1), second coordinate=0×0+3×1=3. · 对于输入(0,1)，第二个坐标=0×0+3×1=3。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- 9260 · Extension · 3.3. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A matrix whose action on a column vector defines a plane transformation. Choose the relationship, show the method, check its assumptions and interpret the result.
将其应用于你的课程
- 9260 · 拓展内容 · 3.3。在分配拓展内容前，请先匹配目标层级和课程大纲要求。
- 在给出最终答案前先展示解题方法，并遵循试卷的计算器及公式使用规则。通过定位第一个无效步骤来复盘错误答案。
一个作用于列向量以定义平面变换的矩阵。选择关系式，展示方法，检查假设并解释结果。