TBIL Activities for Understanding Linear Algebra

Section3.3Image 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 $$\xvec=c_1\vvec_1+c_2\vvec_2+\ldots+c_m\vvec_m\text{.}$$

Activity3.3.0.1.

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

(a)

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

(b)

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

(c)

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 $$\xvec\text{.}$$



(d)

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.
davidaustinm.github.io/ula/sec-jpeg.html#activity-33