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karush

Gold Member

MHB

- 3,269

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nmh{2000}

17.1 Let $T: \Bbb{R}^2 \to \Bbb{R}^2$ be defined by

$$T \begin{bmatrix}

x\\y

\end{bmatrix}

=

\begin{bmatrix}

2x+y\\x-4y

\end{bmatrix}$$

Determine if $T$ is a linear transformation. So if

$$T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})$$

Let $\vec{x}$ and $\vec{y}$ be vectors in $\Bbb{R}^2$ then we can write them as

$$\vec{x}

=\begin{bmatrix}

x_1\\x_2

\end{bmatrix}

, \vec{y}

=\begin{bmatrix}

y_1\\y_2

\end{bmatrix}$$

By definition, we have that

$$T(\vec{x}+\vec{y})

=\begin{bmatrix}

x_1+y_1 \\

x_2+y_2

\end{bmatrix}

=\begin{bmatrix}

2(x_1+y_1)+x_2+y_2\\

x_1+y_1-4(x_2+y_2)

\end{bmatrix}$$

OK just seeing if this is developing as it should

hopefully the next few steps will be an addition property

and this is a linear transformation

17.1 Let $T: \Bbb{R}^2 \to \Bbb{R}^2$ be defined by

$$T \begin{bmatrix}

x\\y

\end{bmatrix}

=

\begin{bmatrix}

2x+y\\x-4y

\end{bmatrix}$$

Determine if $T$ is a linear transformation. So if

$$T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})$$

Let $\vec{x}$ and $\vec{y}$ be vectors in $\Bbb{R}^2$ then we can write them as

$$\vec{x}

=\begin{bmatrix}

x_1\\x_2

\end{bmatrix}

, \vec{y}

=\begin{bmatrix}

y_1\\y_2

\end{bmatrix}$$

By definition, we have that

$$T(\vec{x}+\vec{y})

=\begin{bmatrix}

x_1+y_1 \\

x_2+y_2

\end{bmatrix}

=\begin{bmatrix}

2(x_1+y_1)+x_2+y_2\\

x_1+y_1-4(x_2+y_2)

\end{bmatrix}$$

OK just seeing if this is developing as it should

hopefully the next few steps will be an addition property

and this is a linear transformation

Last edited: