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Matrix Transformations (DP IB Applications & Interpretation (AI))

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  • FrontMatrix Transformations

    What is a transformation matrix?

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  • What is a transformation matrix?

    A transformation matrix is a matrix that is used to determine the coordinates of an image from the transformation of an object.

  • How is a shape on a plane represented by a matrix?

    A shape on a plane can be written as position matrix space open parentheses table row cell x subscript 1 end cell cell x subscript 2 end cell cell x subscript 3 end cell cell... end cell row cell y subscript 1 end cell cell y subscript 2 end cell cell y subscript 3 end cell cell... end cell end table close parentheses, where open parentheses x subscript n comma space y subscript n close parentheses is the coordinate pair of a point on the shape.

  • What is the formula for finding the coordinates of an image under a transformation?

    The formula for finding the coordinates of an image under a transformation is open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row a b row c d end table close parentheses open parentheses table row x row y end table close parentheses plus open parentheses table row e row f end table close parentheses

    Where:

    • open parentheses table row x row y end table close parentheses represents the coordinates of any point in the 2D plane

    • open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses represents the coordinates of the image of point open parentheses table row x row y end table close parentheses

    • open parentheses table row cell table row a b end table end cell row cell table row c d end table end cell end table close parentheses and open parentheses table row e row f end table close parentheses are given matrices

  • How are the coordinates of an original point found from the coordinates of the image of the point and the transformation matrix?

    To find the original points given the image points, the equation open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row a b row c d end table close parentheses open parentheses table row x row y end table close parentheses plus open parentheses table row e row f end table close parentheses can be rearranged to fraction numerator 1 over denominator det space bold italic T end fraction stretchy left parenthesis table row d cell negative b end cell row cell negative c end cell a end table stretchy right parenthesis open square brackets open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses minus open parentheses table row e row f end table close parentheses close square brackets equals stretchy left parenthesis table row x row y end table stretchy right parenthesis.

    You need to find the inverse transformation matrix.

  • True or False?

    The matrix open parentheses table row cell cos theta end cell cell negative sin theta end cell row cell sin theta end cell cell cos theta end cell end table close parentheses represents a clockwise rotation, through an angle of theta, about the point open parentheses 0 comma space 0 close parentheses.

    False.

    The matrix open parentheses table row cell cos theta end cell cell negative sin theta end cell row cell sin theta end cell cell cos theta end cell end table close parentheses does not represent a clockwise rotation, through an angle of theta, about the point open parentheses 0 comma space 0 close parentheses, it represents a counter-clockwise rotation.

    The matrix open parentheses table row cell cos theta end cell cell sin theta end cell row cell negative sin theta end cell cell cos theta end cell end table close parentheses represents a clockwise rotation, through an angle of theta, about the point open parentheses 0 comma space 0 close parentheses.

    These matrices are both given in your exam formula booklet.

  • What matrix represents a reflection in the line y equals left parenthesis tan theta right parenthesis x?

    The matrix open parentheses table row cell cos 2 theta end cell cell sin 2 theta end cell row cell sin 2 theta end cell cell negative cos 2 theta end cell end table close parentheses represents a reflection in the line y equals left parenthesis tan theta right parenthesis x.

    This matrix is given in your exam formula booklet.

  • What transformation does the matrix open parentheses table row k 0 row 0 k end table close parentheses represent?

    The matrix open parentheses table row k 0 row 0 k end table close parentheses represents an enlargement with scale factor k with centre open parentheses 0 comma space 0 close parentheses.

    This matrix is given in your exam formula booklet.

  • True or False?

    The matrix open parentheses table row p row q end table close parentheses represents a translation, of p units in the x-direction and q units in the y-direction.

    True.

    The matrix open parentheses table row p row q end table close parentheses represents a translation, of p units in the x-direction and q units in the y-direction.

    This matrix is not given in your exam formula booklet.

  • What matrix represents a vertical stretch (stretch parallel to the y-axis) with scale factor k?

    The matrix open parentheses table row 1 0 row 0 k end table close parentheses represents a vertical stretch (stretch parallel to the y-axis) with scale factor k.

    This matrix is given in your exam formula booklet.

  • What transformation does the matrix open parentheses table row k 0 row 0 1 end table close parentheses represent?

    The matrix open parentheses table row k 0 row 0 1 end table close parentheses represents a horizontal stretch (stretch parallel to the x-axis) with scale factor k with centre open parentheses 0 comma space 0 close parentheses.

    This matrix is given in your exam formula booklet.

  • True or False?

    A single matrix representing a composite transformation can be found by multiplying together individual transformation matrices.

    True.

    A single matrix representing a composite transformation can be found by multiplying together individual transformation matrices.

    If the transformation represented by matrix bold italic M is applied first, followed by another transformation represented by matrix bold italic N, then the composite matrix is bold italic N bold italic M.

  • Given a transformation represented by the matrix bold italic T, what does bold italic T to the power of 5 represent?

    bold italic T to the power of 5 would be the matrix for five applications of the transformation bold italic T.

  • What does the determinant of a transformation matrix represent?

    The absolute value of the determinant of a transformation matrix is the area scale factor.

    Area scale factor = open vertical bar det bold italic A close vertical bar.

  • How can the determinant of a transformation matrix be used to find the area of an image?

    As the determinant of a transformation matrix is the area scale factor, the product of the area of the original object and the determinant of the transformation matrix will give the area of the image.

    Area of image = open vertical bar det bold italic A close vertical bar × Area of object

  • What will happen to the area of the image if the absolute value of the determinant of the transformation matrix is less than 1.

    If the absolute value of the determinant of the transformation matrix is less than 1, then the area of the image will reduce.

  • What will happen to the orientation of the image if the determinant of the transformation matrix is less than 0.

    If the determinant of the transformation matrix is less than 0, then the orientation of the shape will be reversed.

    I.e. the shape is reflected.

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