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Section 3.3 Image Compression

Suppose that we have a basis \(\bcal=\{\vvec_1,\vvec_2,\ldots,\vvec_m\}\) for \(\real^m\text{.}\) What do we mean by the representation \(\coords{\xvec}{\bcal}\) of a vector \(\xvec\) in the coordinate system defined by \(\bcal\text{?}\)
The components of the vector \(\coords{\xvec}{\bcal}\) are the weights that express \(\xvec\) as a linear combination of the basis vectors; that is, \(\coords{\xvec}{\bcal} = \fourvec{c_1}{c_2}{\vdots}{c_m}\) if \(\xvec=c_1\vvec_1+c_2\vvec_2+\ldots+c_m\vvec_m\text{.}\)


Since we will be using various bases and the coordinate systems they define, let's review how we translate between coordinate systems.


If we are given the representation \(\coords{\xvec}{\bcal}\text{,}\) how can we recover the vector \(\xvec\text{?}\)


If we are given the vector \(\xvec\text{,}\) how can we find \(\coords{\xvec}{\bcal}\text{?}\)


Suppose that
\begin{equation*} \bcal=\left\{\twovec{1}{3},\twovec{1}{1}\right\} \end{equation*}
is a basis for \(\real^2\text{.}\) If \(\coords{\xvec}{\bcal} = \twovec{1}{-2}\text{,}\) find the vector \(\xvec\text{.}\)


If \(\xvec=\twovec{2}{-4}\text{,}\) find \(\coords{\xvec}{\bcal}\text{.}\)
Section 3.3 in Understanding Linear Algebra provides some applications of applications of change of basis, including digital color models and image compression.