1.
Determine whether or not the vector \(\ww=\langle 3,-1, 4, 2\rangle\) in \(\R^4\) belongs to the span of the vectors
\begin{equation*}
\langle 2, 1, 4, -3\rangle, \langle 0, 2, 1, 4\rangle, \langle -1, 1, 0, 2\rangle\text{.}
\end{equation*}
Worksheet 1.5 Worksheet: understanding span
In this worksheet, we will attempt to understand the concept of span. Recall from Section 1.4 that the span of a set of vectors \(\vv_1, \vv_2,\ldots, \vv_k\) in a vector space \(V\) is the set of all linear combinations that can be generated from those vectors.
Therefore, the question “Does the vector \(\ww\) belong to the span of \(\vv_1, \vv_2,\ldots, \vv_k\text{?}\)” is equivalent to asking, “Can I write \(\ww\) as a linear combination of the \(\vv_i\text{?}\)”, which, in turn, is equivalent to asking:
Do there exist scalars \(c_1,c_2,\ldots, c_k\) such that
\begin{equation*}
\ww=c_1\vv_1+c_2\vv_2+\cdots +c_k\vv_k\text{?}
\end{equation*}
In any finite-dimensional vector space, this last question can be turned into a system of equations. If that system has a solution, then yes — your vector is in the span. If the system is inconsistent, then the answer is no.
To assist with solving this problem, a code cell is provided below. Once you have determined the augmented matrix of your system of equations, see Section B.3 for details on how to enter your matrix, and then compute its reduced row-echelon form.
2.
Determine whether or not the polynomial \(q(x) = 4-6x-11x^2\) belongs to the span of the polynomials
\begin{equation*}
p_1(x) = x-3x^2, p_2(x)=2-x, p_3(x) = -1+4x+x^2\text{.}
\end{equation*}
For our next activity, we are going to look at RGB colours. Here, RGB stands for Red, Green, Blue. All colours displayed by your computer monitor can be expressed in terms of these colours.
First, we load some Python libraries we’ll need. These are intended for use in a Jupyter notebook and won’t run properly if you are using Sagecell in the HTML textbook.
Next, we will create a widget that lets us select values for red, green, and blue. The RGB colour system assigns 8-bit values to each colour. Possible values for each range from 0 to 255; this indicates how much of each colour will be blended to create the colour you want. Extensive information on the RGB colour system can be found on wikipedia 1 , and there are a number of good online resources about the use of RGB in web design, such as this one from w3schools 2 .
By moving the sliders generated above, you can create different colours. To see what colour you’ve created by moving the sliders, run the code below.
3.
In what ways can you explain the RGB colour system in terms of span?
4.
Why would it nonetheless be inappropriate to describe the set of all RGB colours as a vector space?
en.wikipedia.org/wiki/RGB_color_modelwww.w3schools.com/colors/colors_rgb.asp
