Plane matrix transformations · Extension
הוצג| English | משמעות |
|---|---|
| transformation matrix/trænsfɔːˈmeɪʃn ˈmeɪtrɪks/ | מטריצת המרה |
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.
The method
- 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.
Why does a zero determinant mean a 2×2 matrix has no inverse?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
The determinant is the signed area scale factor. det=0 collapses a dimension - distinct points share an image - so no inverse transformation can separate them, and division by ad-bc fails.
A worked case
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]].
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
First coordinate=2×1+1×2=4.
Where it goes wrong
- Matrix multiplication is usually not commutative.
- Treat coordinates as column vectors and apply a composition in the correct order.
Not true: The order of a pair of matrix transformations never changes the image. A square matrix with determinant 0 is singular and has no inverse.
Find its y-coordinate.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Second coordinate=0×1+3×2=6.
The order of a pair of matrix transformations never changes the image.
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
A square matrix with determinant 0 is singular and has no inverse.
Applying it
- This 9260 Extension lesson concerns plane transformation matrices.
- Eigenvalues, diagonalisation and characteristic equations are outside its scope.
Find the y-coordinate of A(0,1).
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
For input (0,1), second coordinate=0×0+3×1=3.
Match each matrix concept to its role.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Matrix products act right to left, and an inverse exists exactly when the determinant is nonzero.
In your course
- OxfordAQA International GCSE Mathematics (9260), Extension tier, specification section 3.3.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Transformation matrix: A matrix whose action on a column vector defines a plane transformation.
- Represent a plane transformation using a 2×2 matrix acting on column vectors.
- Watch for: matrix multiplication is usually not commutative.
Write from the image (6, 12)
- The matrix has rows (3, 0) and (1, 2). It acts on (2, 5).
- Find the image. The rightmost transformation acts first when matrices are composed.
The matrix with rows (3, 0) and (1, 2) acts on the point (2, 5). What is the image?
3 times 2 is 6, and 2 plus 10 is 12.
What is the image of (2, 5)?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
The first row gives 6 and the second row gives 12.