FMAT pure1 matrices and transformations Flashcards
What is a matrix?
A matrix is a rectangular array of numbers arranged in rows and columns. It is used to store information.
How are matrices classified?
Matrices are classified by their dimensions (order), written as m × n:
- m: Number of rows,
- n: Number of columns.
What is a square matrix?
A matrix with the same number of rows and columns (e.g., 3 × 3).
What are the conditions for adding or subtracting matrices?
Matrices must have the same dimensions (order) to be added or subtracted.
Example: Add and subtract matrices A and B where:
A = [[2, 3], [-1, 0]], B = [[3, 4], [-1, -2]].
Addition: A + B = [[2 + 3, 3 + 4], [-1 + (-1), 0 + (-2)]] = [[5, 7], [-2, -2]]
Subtraction: A - B = [[2 - 3, 3 - 4], [-1 - (-1), 0 - (-2)]] = [[-1, -1], [0, 2]]
How is scalar multiplication of a matrix performed?
Each element of the matrix is multiplied by the scalar.
Example: For A = [[1, 2], [3, 4]] and scalar 3:
3A = [[3 * 1, 3 * 2], [3 * 3, 3 * 4]] = [[3, 6], [9, 12]]
What is the condition for multiplying two matrices?
Matrix multiplication is possible when the number of columns in the first matrix equals the number of rows in the second matrix.
Example: Multiply matrices A and B where:
A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]
The product AB is:
AB = [[15 + 27, 16 + 28], [35 + 47, 36 + 48]]
= [[19, 22], [43, 50]]
What are the key properties of matrix multiplication?
- Matrix multiplication is not commutative: AB ≠ BA.
- Matrix multiplication is associative: (AB)C = A(BC).
- Matrix multiplication is distributive over addition: A(B + C) = AB + AC.
What is the identity matrix, and how does it affect multiplication?
The identity matrix I is a square matrix with 1s on the diagonal and 0s elsewhere:
I = [[1, 0], [0, 1]].
For any matrix A, IA = AI = A.
Example: Find the product of A and the identity matrix where:
A = [[2, 3], [4, 5]], I = [[1, 0], [0, 1]].
AI = [[21 + 30, 20 + 31], [41 + 50, 40 + 51]]
= [[2, 3], [4, 5]]
AI = A
What is a linear transformation in two dimensions?
A linear transformation maps a point (x, y) to (x’, y’) using:
x’ = ax + by
y’ = cx + dy
This can be written as a matrix equation:
[[x’], [y’]] = [[a, b], [c, d]] × [[x], [y]]
How can you find the image of a shape under a transformation matrix?
Write the points of the shape as a matrix, multiply by the transformation matrix, and solve for the resulting points.
Example: Find the image of A(3, 1), B(-2, 4), C(5, -1) under the transformation matrix [[2, 3], [-1, 0]].
- Points matrix: [[3, -2, 5], [1, 4, -1]]
- Multiply by the transformation matrix:
[[2, 3], [-1, 0]] × [[3, -2, 5], [1, 4, -1]] - Result:
[[9, 8, 7], [-3, 2, -5]] - Image:
A → (9, -3), B → (8, 2), C → (7, -5)
What is the transformation matrix for a stretch parallel to the x-axis?
[[k, 0], [0, 1]]
scale factor k
What is the transformation matrix for a stretch parallel to the y-axis?
[[1, 0], [0, k]]
scale factor k
What is the transformation matrix for an enlargement centred at the origin?
[[k, 0], [0, k]]
scale factor k
What is the matrix for a 90° anticlockwise rotation about the origin?
[[0, -1], [1, 0]]
What is the matrix for a 90° clockwise rotation about the origin?
[[0, 1], [-1, 0]]
What is the general matrix for an anticlockwise rotation through an angle θ?
[[cos(θ), -sin(θ)], [sin(θ), cos(θ)]]
Example: Find the matrix for an anticlockwise rotation through 140° about the origin.
- Use the general rotation matrix:
[[cos(140°), -sin(140°)], [sin(140°), cos(140°)]] - Approximate values:
[[-0.766, -0.642], [0.642, -0.766]]
What is the matrix for a reflection in the x-axis?
[[1, 0], [0, -1]]
What is the matrix for a reflection in the y-axis?
[[-1, 0], [0, 1]]
What is the matrix for a reflection in the line y = x?
[[0, 1], [1, 0]]
What is the matrix for a reflection in the line y = -x?
[[0, -1], [-1, 0]]
What is the matrix for a shear parallel to the x-axis with factor k?
[[1, k], [0, 1]]
What is the matrix for a shear parallel to the y-axis with factor k?
[[1, 0], [k, 1]]
How are combined transformations represented using matrices?
The combined transformation is represented by multiplying the matrices of the individual transformations. Order matters: Apply the rightmost transformation first.
Example: Find the matrix for a reflection in the x-axis followed by a 90° anticlockwise rotation.
- Reflection in x-axis: [[1, 0], [0, -1]].
- 90° rotation: [[0, -1], [1, 0]].
- Combined: [[0, -1], [1, 0]] × [[1, 0], [0, -1]] = [[0, 1], [1, 0]].
This represents a reflection in the line y = x.
What is an invariant point in a transformation?
An invariant point is a point that is mapped to itself by the transformation.
What is the condition for finding invariant points in a 2x2 transformation matrix?
Solve the matrix equation:
[[a, c], [b, d]] * [[x], [y]] = [[x], [y]].
This results in two equations:
1. ax + cy = x,
2. bx + dy = y.
What is the significance of invariant points in transformations?
For linear transformations:
- The origin is either the only invariant point.
- Or all invariant points lie on a straight line through the origin.
Finding invariant points can also help determine properties like the mirror line in a reflection.
Example: Find the line of invariant points for the matrix [[2, 1], [2, 3]].
- Solve:
[[2, 1], [2, 3]] * [[x], [y]] = [[x], [y]].
This expands to:
2x + y = x and 2x + 3y = y. - Simplify:
x + y = 0 (both equations are equivalent). - Solution:
The line of invariant points is y = -x.
What is an invariant line in a transformation?
An invariant line is a line that maps to itself under the transformation, but the points on the line are not necessarily invariant points.
How do you find invariant lines for a 2x2 transformation matrix?
- Assume the invariant line is y = mx + c.
- Substitute a general point (x, mx + c) into the transformation matrix.
- Solve for m and c by ensuring the image satisfies y = mx + c.
What is the relationship between invariant points and invariant lines?
- A line of invariant points is also an invariant line.
- An invariant line is not necessarily a line of invariant points.
Example: Find the invariant lines for the matrix [[2, 1], [2, 3]].
- Assume y = mx + c.
- Apply the transformation:
[[2, 1], [2, 3]] * [[x], [mx + c]].
This expands to:
[[2x + mx + c], [2x + 3mx + 3c]]. - Ensure the image satisfies y = mx + c:
2m = m or m = -1.
If m = 2, c can be any value, giving lines y = 2x + c.
If m = -1, c must be 0, giving the line y = -x.
Example: Find the line of invariant points and invariant lines for the matrix [[2, 1], [2, 3]].
- Line of invariant points: y = -x.
- Invariant lines: y = 2x + c (any c) and y = -x.
Why is identifying invariant points and lines useful in transformations?
They provide insights into the geometric effects of transformations, such as rotations, reflections, or stretches.
