Update main.tex

Author: annadavismath

LTR-0030/main.tex

@@ -205,7 +205,7 @@ \subsection*{Linear Transformations of $\RR^n$ and the Standard Matrix of the In
 Let $$V=\text{span}\left(\begin{bmatrix}1\\0\\0\end{bmatrix}, \begin{bmatrix}1\\1\\1\end{bmatrix}\right)$$
 Define a linear transformation $$T:V\rightarrow \RR^2$$
 by $$T\left(\begin{bmatrix}1\\0\\0\end{bmatrix}\right)=\begin{bmatrix}1\\1\end{bmatrix}\quad \text{and} \quad T\left(\begin{bmatrix}1\\1\\1\end{bmatrix}\right)=\begin{bmatrix}0\\1\end{bmatrix}$$
-Observe that $\left\{\begin{bmatrix}1\\0\\0\end{bmatrix}, \begin{bmatrix}1\\1\\1\end{bmatrix}\right\}$ is a basis of $V$ (why?). The information about the images of the basis vectors is sufficient to define a linear transformation.  This is because every vector $\vec{v}$ in $V$ can be expressed as a linear combination of the basis elements.  The image, $T(\vec{v})$, can be found by applying the linearity properties.
+Observe that $\left\{\begin{bmatrix}1\\0\\0\end{bmatrix}, \begin{bmatrix}1\\1\\1\end{bmatrix}\right\}$ is a basis of $V$ (why?). The information about the images of the basis vectors is sufficient to define a linear transformation.  This is because every vector $\vec{v}$ in $V$ can be expressed as a linear combination of the basis elements in a unique way.  The image, $T(\vec{v})$, can be found by applying the linearity properties.
 At this point we know what transformation $T$ does, but it is still unclear what the matrix of this linear transformation is.
 
 Geometrically speaking, the domain of $T$ is a plane in $\RR^3$ and its codomain is $\RR^2$.