# Homework: 13s: Critical Thinking Challenge

• MHB
• karush
In summary, the conversation involves a discussion of a homework problem and the steps to solve it. It includes a matrix transformation S and T, and the process of applying them in a specific order to the matrix. The final solution is a simplified matrix after applying the transformations.
karush
Gold Member
MHB
View attachment 8848

ok just sent this homework in (maybe typos}
on TS i think this is as far you can go with the current dimensions

critical thinking accepted...

karush said:
ok just sent this homework in (maybe typos}
on TS i think this is as far you can go with the current dimensions

critical thinking accepted...
The first two are right, though I'm really not fond of your text trying to add a 2x1 matrix with a 3x1 in the second one. That's rather illegal, but not your problem.

As for $$\displaystyle S^3$$, take it step by step.
$$\displaystyle S \left [ \begin{matrix} x \\ y \end{matrix} \right ] = \left [ \begin{matrix} x - 2y \\ 3x - y \end{matrix}\right ]$$

$$\displaystyle S^2 \left [ \begin{matrix} x \\ y \end{matrix} \right ] = S \left ( S \left [ \begin{matrix} x \\ y \end{matrix} \right ] \right )$$

$$\displaystyle = S \left [ \begin{matrix} x - 2y \\ 3x - y \end{matrix} \right ] = \left [ \begin{matrix} (x - 2y) - 2(3x - y) \\ 3(x - 2y) - (3x -y) \end{matrix} \right ] = \left [ \begin{matrix} -5x \\ -5y \end{matrix} \right ]$$

Now you finish the rest.

-Dan

karush said:
$\displaystyle = S \left [ \begin{matrix} x - 2y \\ 3x - y \end{matrix} \right ] = \left [ \begin{matrix} (x - 2y) - 2(3x - y) \\ 3(x - 2y) - (3x -y) \end{matrix} \right ] = \left [ \begin{matrix} -5x \\ -5y \end{matrix} \right ]$so then kinda maybe
$S\left [ \begin{matrix} -5x \\ -5y \end{matrix} \right ] =\left [ \begin{matrix} -5(x-2y) \\ -5(3x-y) \end{matrix} \right ] =\left [ \begin{matrix} -5x-10y \\ -15x-5y \end{matrix} \right ]$
The transformation S takes a value $$\displaystyle x \to x - 2y$$ and $$\displaystyle y \to 3x - y$$, so your new x value will be (-5x) - 2(-5y) and your new y value will be 3(-5x) - (-5y).

-Dan

so you mean this
$S\left [ \begin{matrix} -5x \\ -5y \end{matrix} \right ] =\left [ \begin{matrix} (-5x) - 2(-5y) \\ 3(-5x) - (-5y) \end{matrix} \right ] =\left [ \begin{matrix} -5x-10y \\ -15x-5y \end{matrix} \right ]$

so for ST if
$S\left(\left[\begin{array}{c}x \\ y \end{array}\right]\right) =\left[\begin{array}{c}x-2y \\ 3x-y \end{array}\right], \quad T\left(\left[\begin{array}{c}x \\ y \end{array}\right]\right) =\left[\begin{array}{c}x+y \\ x-y\\2x+3y \end{array}\right]$

then

$S\left(\left[\begin{array}{c}x-2y \\ 3x-y \end{array}\right]\right) =\left[\begin{array}{c}(x-2y)+(3x-y) \\ (x-2y)-(3x-y)\\2(x-2y)+3(3x-y) \end{array}\right]$
if ok then simplify..

Last edited:
karush said:
so you mean this
$S\left [ \begin{matrix} -5x \\ -5y \end{matrix} \right ] =\left [ \begin{matrix} (-5x) - 2(-5y) \\ 3(-5x) - (-5y) \end{matrix} \right ] =\left [ \begin{matrix} -5x-10y \\ -15x-5y \end{matrix} \right ]$
Watch the double negatives!

-Dan

- - - Updated - - -

karush said:
so for ST if
$S\left(\left[\begin{array}{c}x \\ y \end{array}\right]\right) =\left[\begin{array}{c}x-2y \\ 3x-y \end{array}\right], \quad T\left(\left[\begin{array}{c}x \\ y \end{array}\right]\right) =\left[\begin{array}{c}x+y \\ x-y\\2x+3y \end{array}\right]$

then

$S\left(\left[\begin{array}{c}x-2y \\ 3x-y \end{array}\right]\right) =\left[\begin{array}{c}(x-2y)+(3x-y) \\ (x-2y)-(3x-y)\\2(x-2y)+3(3x-y) \end{array}\right]$
if ok then simplify..
I think you mean
$$\displaystyle T \left ( \left [ \begin{array}{c}x-2y \\ 3x-y \end{array} \right ] \right )$$
in that last line. Otherwise, yes, good!

-Dan

A random thought. When you were doing the TS part you do realize you do S and then T, right?

-Dan

topsquark said:
A random thought. When you were doing the TS part you do realize you do S and then T, right?

-Dan
Yeah I thot it was a product
Not a composite

## 1. What is the purpose of the "Homework: 13s: Critical Thinking Challenge" assignment?

The purpose of this assignment is to challenge students to think critically and creatively about a given topic or problem. It aims to develop their problem-solving skills and encourage them to approach tasks in a more analytical and strategic manner.

## 2. How does the "Homework: 13s: Critical Thinking Challenge" help students in their academic and professional lives?

This assignment helps students develop important skills such as critical thinking, problem-solving, and analytical reasoning, which are highly valued in both academic and professional settings. These skills can also help students excel in their future studies and careers.

## 3. What are some examples of critical thinking skills that students can develop through this assignment?

Some examples of critical thinking skills that students can develop through this assignment include logical reasoning, evaluating evidence, identifying biases, and making informed decisions. Students may also improve their creativity, communication, and collaboration skills.

## 4. How can teachers support students in completing the "Homework: 13s: Critical Thinking Challenge" successfully?

Teachers can support students by providing clear instructions and expectations for the assignment, offering guidance and feedback throughout the process, and creating a supportive and inclusive learning environment. They can also provide additional resources and examples to help students better understand the critical thinking process.

## 5. Can students work in groups for the "Homework: 13s: Critical Thinking Challenge"?

Yes, students can work in groups for this assignment. Collaborative learning can enhance critical thinking skills by allowing students to share ideas, perspectives, and strategies. However, it is important for each student to contribute and engage in the critical thinking process, rather than relying on others to do the work for them.

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