Intereting Posts

A classic exponential inequality: $x^y+y^x>1$
Prove that the degree of the splitting field of $x^p-1$ is $p-1$ if $p$ is prime
Sum $\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$
Limit of the sequence $\{n^n/n!\}$, is this sequence bounded, convergent and eventually monotonic?
Tensors, what should I learn before?
Prove that: $\sin{\frac{\pi}{n}} \sin{\frac{2\pi}{n}} …\sin{\frac{(n-1)\pi}{n}} =\frac{n}{2^{n-1}}$
$ S_n=\sum_{k=1}^n\frac{1}{k}$ then is $S_n$ bounded?
Relationship between Dixonian elliptic functions and Borwein cubic theta functions
Entire function with vanishing derivatives?
What is the condensation set of a fractal?
Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?
The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the point on the curve farthest from the line $x = y$
“Too simple to be true”
Find and classify the bifurcations that occur as $\mu$ varies for the system
Convergence of the sequence $\frac{1}{n\sin(n)}$

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?

- Operators on a Tensor Product Space
- There exists a vector $c\in C$ with $c\cdot b=1$
- Cauchy-Schwarz in complex case, using discriminant
- Is $(\mathbf{V} \cap \mathbf{W})^{\bot}=(\mathbf{V}^{\bot} \cap \mathbf{W}^{\bot})$?
- Does $\det(A) \neq 0$ (where A is the coefficient matrix) $\rightarrow$ a basis in vector spaces other than $R^{n}$?
- Do there exist vector spaces over a finite field that have a dot product?
- What is the agreed upon definition of a “positive definite matrix”?
- Transpose of a linear mapping
- Should I use sets or tuples when dealing with linear dependence?
- Vector dimension of a set of functions

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 $a + 0b + 0c + d = 0$.

- In the card game Set, what's the probability of a Set existing in n cards?
- How to understand Cauchy's proof of AM-GM inequality(the last step)
- Show that $f$ is uniformly continuous if limit exists
- Can I switch to polar coordinates if my function has complex poles?
- Cube of harmonic mean
- Minimum degree of a graph and existence of perfect matching
- How to evaluate $\int_0^{2\pi} \frac{d\theta}{A+B\cos\theta}$?
- Proof for Inequality
- Does $f$ monotone and $f\in L_{1}([a,\infty))$ imply $\lim_{t\to\infty} t f(t)=0$?
- Is the series $\sum \sin^n(n)$ divergent?
- Find limit $a_{n + 1} = \int_{0}^{a_n}(1 + \frac{1}{4} \cos^{2n + 1} t)dt,$
- Convergence of $\sum_{n=1}^\infty\frac{1}{2\cdot n}$
- Count the number of topological sorts for poset (A|)?
- Why $\sqrt{{2 + \sqrt 5 }} + \sqrt{{2 – \sqrt 5 }}$ is a rational number?
- Entire “periodic” function