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
Worksheet
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
\(f(x)=\dfrac{a}{x}+b,\,\,a,b\in\R\)
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 \(y=x^3\)
Got questions? Share and ask here...
Google 
Facebook