Qucik help with transformation matrices

[rock.freak667]

Homework Statement

Three transformations of the x-y plane are defined as follows.
$$T_1$$: enlargement with centre O(the origin) and scale factor 5
$$T_2$$: Anti-clockwise rotation about the origin O through an angle $$tan^{-1}(\frac{4}{3})$$
$$T_3$$: A stretch parallel to the x-axis(with the y-axis invariant) with scale factor 2.

The transformation $$T_4$$ is the result of applying $$T_1,T_2,T_3$$ in that order. Find the matrix which represents $$T_4$$

Homework Equations

The Attempt at a Solution

$T_1 =\left(<br /> \begin{array}{cc}<br /> 5 & 0\\<br /> 0 & 5<br /> \end{array}<br /> \right)$


$T_2 =\left(<br /> \begin{array}{cc}<br /> cos(tan^{-1}(\frac{4}{3})) & -sin(tan^{-1}(\frac{4}{3}))\\<br /> sin(tan^{-1}(\frac{4}{3})) & cos(tan^{-1}(\frac{4}{3}))<br /> \end{array}<br /> \right)$


$$T_3 =\left(<br /> \begin{array}{cc}<br /> 2 & 0\\<br /> 0 & 1<br /> \end{array}<br /> \right)$$

and $$T_4 = T_3*T_2*T_1$$
Is the matrices I put correct and is $$T_4$$ correct?

 

[D H]

Other than T₂ (which is correct, but should be simplified), you are fine. Transformation matrices chain right-to-left.

 

[rock.freak667]

Other than T₂ (which is correct, but should be simplified), you are fine. Transformation matrices chain right-to-left.

so then $$T_4$$ is just to multiply the transformations in the order given ?

 

[HallsofIvy]

Yes, that's the whole point- applying the transformation corresponds to multiplying by the matrix.