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
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