Intereting Posts

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use of $\sum $ for uncountable indexing set
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Golbach's partitions: is there always one common prime in $G(n)$ and $G(n+6)$ , $n \ge 8$ (or a counterexample)?
Number of times the ball bounces back when dropped from a building of height h

I have a question that I do not understand and it goes like this:

Find a basis for the set $W$ of all matrices A in $M_{2\times2}$ with trace $0$: i.e. all matrices

$$

\begin{pmatrix}

a & b\\

c & d \

\end{pmatrix}

$$

with $a+d = 0$.

What is the dimension W?

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So you really have the set of matrices of the form

$$

W = \{\pmatrix{a & b \\ c & -a}\}

$$

I claim that a basis is

$$

e_1 = \pmatrix{1 & 0 \\ 0 & -1}, \\

e_2 = \pmatrix{0 & 1 \\ 0 & 0},\\

e_3 = \pmatrix{0 & 0 \\ 1 & 0}.

$$

All you have to do is to prove that $e_1, e_2, e_3$ span all of $W$ and that they are linearly independent.

I will let you think about the spanning property and show you how to get started with showing that they are linearly independent. Assume that

$$

ae_1 + be_2 + ce_3 = 0.

$$

This means that

$$

\pmatrix{a & b \\ c & -a} = \pmatrix{0 & 0 \\ 0 & 0},

$$

and so $a = b = c = 0$. Hence we have linear independence.

Any required matrix of size 2×2 can be represented as a linear combination of

\begin{align}

\begin{bmatrix}

0&1\\0&0

\end{bmatrix}\\

\begin{bmatrix}

0&0\\1&0

\end{bmatrix}\\

\begin{bmatrix}

1&0\\0&-1

\end{bmatrix}\\

\end{align}

Dimension is 3.

Alternatively, $\operatorname{trace}\begin{pmatrix}a&b\\c&d\end{pmatrix}=0$ if and only if $(a,b,c,d)^T\perp(1,0,0,1)^T$ in $\mathbb{R}^4$.

Maybe it would help to forget the context and focus on the algebraic problem:

Find all solutions for $(a,b,c,d)$ to the linear system of one equation in four variables $a+d=0$. Write down a basis for the solution space. What is its dimension?

If you have trouble dealing with degeneracy (e.g. you think “but $b$ and $c$ aren’t in the equation!”), then as a temporary measure until you become more comfortable with it, it may help to write it as $a + 0b + 0c + d = 0$.

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