Plane matrix transformations · Extension
| English | 中文 | Pinyin |
|---|---|---|
| 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.
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.
Which description correctly defines transformation matrix?
A matrix whose action on a column vector defines a plane transformation.
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.
Plane matrix transformations
Represent a plane transformation using a 2×2 matrix acting on column vectors
Compare the model with the worked case and explain one change.
Find the x-coordinate of A(1,2) for A=[[2,1],[0,3]].
First coordinate=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.
Find its y-coordinate.
Second coordinate=0×1+3×2=6.
The order of a pair of matrix transformations never changes the image.
Matrix multiplication is usually not commutative. Treat coordinates as column vectors and apply a composition in the correct order.
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.
Find the y-coordinate of A(0,1).
For input (0,1), second coordinate=0×0+3×1=3.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
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.