# -307.17.1 Show that S and T are both linear transformations

• MHB
• karush
In summary, the conversation discusses a problem from an overleaf homework assignment and the need to prove the linearity of two transformations, S and T. The conversation includes an example and a discussion on how to show the preservation of addition and scalar multiplication for S and T. The conversation also covers finding the matrices of S and T with respect to the standard basis for $\Bbb{R}^2$.
karush
Gold Member
MHB
View attachment 8994

ok this is a clip from my overleaf homework reviewing

just seeing if I am going in the right direction with this

their was an example to follow but it also was a very different problem

much mahalo

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The last two lines have two equalities. Their status is not clear. Please say for each equality if it is something you plan to prove, something you assume, or something you have proved and how (by definition, by laws of algebra, etc.).

The claim that $S$ is a linear transformation requires a proof, and a proof is not simply some collection of formulas. A proof is an argument that starts with assumptions and arrives and the desires conclusion. proofs are best expressed using text in a natural language (English) written in complete grammatical sentences.

Evgeny.Makarov said:
The last two lines have two equalities. Their status is not clear. Please say for each equality if it is something you plan to prove, something you assume, or something you have proved and how (by definition, by laws of algebra, etc.).

The claim that $S$ is a linear transformation requires a proof, and a proof is not simply some collection of formulas. A proof is an argument that starts with assumptions and arrives and the desires conclusion. proofs are best expressed using text in a natural language (English) written in complete grammatical sentences.
View attachment 9009
ok here is the example I am trying to follow

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Yes, so far what you wrote is correct, and it follows the example.

Evgeny.Makarov said:
Yes, so far what you wrote is correct, and it follows the example.
Let
$S:\Bbb{R}^2\to \Bbb{R}^2$ and $T:\Bbb{R}^2 \to \Bbb{R}^2$ be transformations defined by
$\begin{bmatrix} x\\y \end{bmatrix}= \begin{bmatrix} 2x+y \\ x-y \end{bmatrix}, \quad T \begin{bmatrix}x\\y \end{bmatrix}= \begin{bmatrix}x-4y\\3x \end{bmatrix}$
Show that S and T are both linear transformations
\begin{align*}\displaystyle S\left(\left[\begin{array}{} x_2 \\ y_2 \end{array}\right] +\left[\begin{array}{} x_2\\y_2\end{array}\right]\right) &=S\left[\begin{array}{}x_1+x_2\\y_1+y_2\end{array}\right]\\ &=\left[\begin{array}{c}2(x_1+x_2)+(y_1+y_2) \\ (x_1+x_2)-(y_1+y_2) \end{array}\right]\\ &=\left[\begin{array}{c} 2x_1+2x_2\\x_1+x_2 \end{array}\right] +\left[\begin{array}{c}y_1+y_2\\-y_1-y_2) \end{array}\right] \end{align*}
ok for some reason I can't see how this is going to preserve addition
or is there another way to show transformaton?

The last line should be

$$\displaystyle \begin{bmatrix}2x_1+y_1\\x_1-y_1\end{bmatrix}+\begin{bmatrix}2x_2+y_2\\x_2-y_2\end{bmatrix}$$.

ok here is the whole story... typo's maybe
$S:\Bbb{R}^2\to \Bbb{R}^2$ and $T:\Bbb{R}^2 \to \Bbb{R}^2$ be transformations defined by
$\begin{bmatrix}x\\y \end{bmatrix}= \begin{bmatrix}2x+y \\x-y \end{bmatrix}, \quad T\begin{bmatrix}x\\y \end{bmatrix}= \begin{bmatrix}x-4y\\3x \end{bmatrix}$
Show that S and T are both linear transformations
\begin{align*}\displaystyle S\left(\left[\begin{array}{} x_2 \\ y_2 \end{array}\right] +\left[\begin{array}{} x_2\\y_2\end{array}\right]\right) &=S\left[\begin{array}{}x_1+x_2\\y_1+y_2\end{array}\right]\\ &=\left[\begin{array}{c}2(x_1+x_2)+(y_1+y_2) \\ (x_1+x_2)-(y_1+y_2) \end{array}\right]\\ &=\begin{bmatrix}2x_1+y_1\\x_1-y_1\end{bmatrix}+\begin{bmatrix}2x_2+y_2\\x_2-y_2\end{bmatrix}\\ &=S \begin{bmatrix}x_1\\y_1\end{bmatrix} +S\begin{bmatrix} x_2\\y_2\end{bmatrix}\end{align*}
S preserves addition, If c is any scalar.
$S\left(c\begin{bmatrix} x_1\\y_1\end{bmatrix}\right) =S\begin{bmatrix} cx_2\\cy_2 \end{bmatrix} =\begin{bmatrix} 2cx+cy \\ cx-cy \end{bmatrix} =c\begin{bmatrix} 2x+y \\ x-y \end{bmatrix} =cS\begin{bmatrix} x_1\\y_1\end{bmatrix}$
and consequently T preserves scalar multiplication.

ok (b) and (c) came with this problem, but I think I got them ok but wanted to post it.(b) Find $ST \begin{bmatrix} x\\y \end{bmatrix}$
$$ST\begin{bmatrix}x\\y\end{bmatrix} =S\left(T\begin{bmatrix} x-4y\\3x \end{bmatrix}\right) =\left[\begin{array}{c} 2(x-4y)+3x \\ x-4y-3x \end{array}\right]$$
and $T^2 \begin{bmatrix} x\\y \end{bmatrix}$
$$T^2\left(\left[\begin{array}{c} x \\ y \end{array} \right]\right) =T\left(T\left(\left[\begin{array}{c} x \\ y \end{array}\right]\right)\right) =T\left(\left[\begin{array}{c} x-4y\\3x \end{array}\right]\right) =\left[\begin{array}{c} x-4y-4(3x) \\ 3(x-4y) \end{array}\right]$$
(c) Find the matrices of S and T with respect to the standard basis for $\Bbb{R}^2$.
$$\displaystyle\left[S\right]_\infty^\infty =\left[\begin{array}{cc} 2&1\\1&-1 \end{array}\right], \quad \left[T\right]_\infty^\infty =\left[\begin{array}{cc} 1&-4\\3&0 \end{array}\right]$$

## What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the algebraic structure of the original space. In simpler terms, it is a function that takes in a vector and outputs another vector, while maintaining certain properties such as linearity and preservation of operations.

## What does it mean to show that S and T are both linear transformations?

Showing that S and T are both linear transformations means proving that they both meet the criteria for being considered linear transformations. This includes satisfying properties such as preservation of vector addition and scalar multiplication, as well as being defined over a vector space.

## How do you prove that S and T are both linear transformations?

To prove that S and T are both linear transformations, you must show that they satisfy the two main properties of linearity: preservation of vector addition and preservation of scalar multiplication. This can be done by using mathematical techniques such as substitution and direct proof.

## Why is it important to show that S and T are both linear transformations?

It is important to show that S and T are both linear transformations because it confirms that they adhere to the fundamental properties of linear transformations. This allows us to use them in further mathematical calculations and proofs, knowing that they will behave in a predictable and consistent manner.

## What are some real-world applications of linear transformations?

Linear transformations have many real-world applications, including image and signal processing, computer graphics, and data compression. They are also used in physics, engineering, and economics to model and analyze various systems and processes.

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