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Matrices and Linear Transformations

This tutorial covers material encountered in chapters 2 and 3 of the VCE Mathematical Methods Textbook, namely:

  • Simultaneous equations
  • Matrices and their components
  • Algebra of matrices
  • Linear transformations of functions
  • Linear transformations via matrices

Q1 – Algebra of Matrices

Q2 – Solving Simultaneous Linear Equations

Q3 – Linear Transformation via Matrices

Q5 – Transformation: Reflection, Dilation and Translation

Q6 – Combination of Transformations


Q1. For the matrices A=\begin{bmatrix} 1&2\\3&4 \end{bmatrix},\, B=\begin{bmatrix} 1&0\\-1&1 \end{bmatrix},\, C=\begin{bmatrix} 2\\2 \end{bmatrix} find:

(a) 2A

(b) A+B

(c) AB

(d) BA

(e) BC

(f) 2AB-3BA

Q2. Solve the following simultaneous linear equations (A real parameter may be necessary):

(a) \begin{cases} 3x+5y+2z=8 \\ -3x-5y-4z=16 \end{cases}

(b) \begin{cases} 2x-z+3y=9 \\3x+z=3 \end{cases}

Q3. Consider the transformation T:\R^2 \to \R^2 defined by:

T\left(\begin{bmatrix} x\\y \end{bmatrix}\right)=\begin{bmatrix} 1&0\\0&-3 \end{bmatrix} \begin{bmatrix} x\\y \end{bmatrix} + \begin{bmatrix} -2\\3 \end{bmatrix}

Find the image of the functions f(x)=x^2 and g(x)=3\sqrt{x-2}+2 under this transformation.

Q4. If the function f:\R\to\R has the form


and passes through the points (2,-1) and (4,4) find a and b.

Q5. Find the rule for the image of the graph y=e^x under the following sequence of transformations:

(i) reflection in the x-axis.

(ii) dilation by factor 3 from the y-axis.

(iii) translation of 2 units in the negative x-axis and 3 units in the positive y-axis.

Q6. Find a sequence of transformations that takes the graph of y=3(x-1)^3+5 to the graph of y=x^3

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