Determining If a Function is a Linear Transformation

One consequence of the definition of a linear transformation is that every linear transformation must satisfy $$ T(0_V)=0_W $$ where $0_V$ and $0_W$ are the zero vectors in $V$ and $W$, respectively. Therefore any function for which $T(0_V)\neq 0_W$ cannot be a linear transformation.

In your second example, $$ T\Big(\begin{bmatrix}0\\0\end{bmatrix}\Big)=\begin{bmatrix}0\\1\end{bmatrix}\neq\begin{bmatrix}0\\0\end{bmatrix}$$ so this tells you right away that $T$ isn't linear.


This is not a linear transformation.

Indeed,

$T(1,0)+T(1,0)=(1,1)+(1,1)=(2,2)$

On the other hand

$T(2,0)=(2,1)\neq (2,2)$

For T to be linear, these would have to be equal.

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Linear Algebra

Linear Transformations

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