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Matrices and Transformation

VECTORS
  • A vector quantity has both magnitude and direction. For example:
    • Vectors a and b represented by the line segments can be added using the parallelogram rule or the nose-to-tail method
  • Multiplication by a scalar:
    • A scalar quantity has a magnitude but no direction.
    • The negative sign reverses the direction of the vector.
  • Column vector:
    • The top number is the horizontal component and the bottom component is the vertical component.
  • Parallel vectors:
    • Vectors are parallel if they have the same direction.
    • In general, the vector is parallel to
  • Modulus of a vector:
    • In general, if
MATRICES
  • Addition:
  • Multiplication by scalar:
  • Multiplication by vector:
    • You can only multiply if the number of columns on the left matrix is equal to the number of rows on the right matrix.
  • Determinant:
    • Determinant = leading diagonal – secondary diagonal
  • Inverse:
    • To work out the inverse of a matrix, alternate the leading diagonal’s elements, negate the secondary diagonal, and multiply by 1/determinant
TRANSFORMATION
  • Reflection (M):
    • When describing a reflection, the position of the mirror line is essential.
  • Rotation (R):
    • To describe a rotation, the center of rotation, the angle of rotation, and the direction of the rotation are required.
    • A clockwise rotation is negative and an anticlockwise rotation is positive.
  • Translation (T):
    • When describing a translation, it is necessary to give the translation vector
  • Enlargement (E):
    • To describe an enlargement, state the scale factor,k, and the center of enlargement.
    • If k > 0, both object and image lie on the same side of the center of enlargement.
    • If k < 0, object and image lie on opposite sides of the center of enlargement.
TRANSFORMATION BY MATRICES
  • Reflection:
    • — Reflection in the x – axis
    • — Reflection in the y – axis
    • — Reflection in the line y = -x
  • Enlargement:
    • — where k = scale factor and center of enlargement = (0, 0)
  • Rotation:
    • — Rotation 90° anticlockwise, center (0, 0)
    • — Rotation 90° clockwise, center (0, 0)
    • — Rotation 180° clockwise/anticlockwise, center (0, 0)
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