14-240/Tutorial-October14
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Boris
Elementary and (Not So Elementary) Errors in Homework
(1) Let [math]\displaystyle{ M_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, M_2 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \\ \end{pmatrix}, M_3 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix} }[/math] be matrices.
We want to equate [math]\displaystyle{ span(M_1, M_2, M_3) }[/math] to the set of all symmetric [math]\displaystyle{ 2 \times 2 }[/math] matrices. Here is the wrong way to write this:
[math]\displaystyle{
span(M_1, M_2, M_3) =
\begin{pmatrix}
a & b \\
b & c \\
\end{pmatrix}
}[/math].
Firstly, [math]\displaystyle{ span(M_1, M_2, M_2) }[/math] is the set of all linear combinations of [math]\displaystyle{ M_1, M_2, M_3 }[/math]. To equate it to a single symmetric [math]\displaystyle{ 2 \times 2 }[/math] matrix makes no sense. Secondly, the elements [math]\displaystyle{ a, b, c, d }[/math] are undefined. What are they suppose to represent? Rational numbers? Real numbers? Members of the field of two elements? The following way of writing erases those issues.
[math]\displaystyle{
span(M_1, M_2, M_3) = \{
\begin{pmatrix}
a & b \\
b & c \\
\end{pmatrix}
:a, b, c \in F \}
}[/math] where [math]\displaystyle{ F }[/math] is an arbitrary field.
