Intereting Posts

Examples where it is easier to prove more than less
Computing $\sum_{m \neq n} \frac{1}{n^2-m^2}$
Group with order $p^2$ must be abelian . How to prove that?
Symmetric vs. Positive semidefinite
Finite Sum $\sum_{i=1}^n\frac i {2^i}$
Infinitely many systems of $23$ consecutive integers
Check my workings: Show that $\lim_{h\to0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f''(x)$
The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$
How to find Matrix Inverse over finite field?
Rotations and the parallel postulate.
A smooth function $f:S^1\times S^1\to \mathbb R$ must have more than two critical points.
Removable singularity and laurent series
Example of a non-separable normal extension
Dirichlet's Divisor Problem
Integral change that I don't understand

I’m sorry if this is a silly question. I’m new to the notion of bases and all the examples I’ve dealt with before have involved sets of vectors containing real numbers. This has led me to assume that bases, by definition, are made up of a number of $n$-tuples.

However, now I’ve been thinking about a basis for all $n\times n$ matrices and I keep coming back to the idea that the simplest basis would be $n^2$ matrices, each with a single $1$ in a unique position.

Is this a valid basis? Or should I be trying to get column vectors on their own somehow?

- Can the product of an $4\times 3$ matrix and a $3\times 4$ matrix be invertible?
- How to show that $\det(AB) =\det(A)\det(B)$
- Eigenvalue Problem — prove eigenvalue for $A^2 + I$
- Interpreting the Cayley-Hamilton theorem
- What are the “building blocks” of a vector?
- What's the meaning of the transpose?

- Generic topology on a vector space?
- Why is $1, (x-5)^2, (x-5)^3$ a basis of $U=\{p \in \mathcal P_3(\mathbb R) \mid p'(5)=0\}$?
- existence of a complementary subspace
- Is $SO_n({\mathbb R})$ a divisible group?
- Hessian matrix of a quadratic form
- Determinant of a special skew-symmetric matrix
- Algebraically-nice general solution for last step of Gaussian elimination to Smith Normal Form?
- Angle preserving linear maps
- eigen decomposition of an interesting matrix
- GRE linear algebra question

Elements of a basis of a vector space always have to be elements of the vector space in the first place. Hence, if you are looking for a basis of the space of all $n\times n$ matrices, then matrices actually **are** your vectors and the only choice for what a basis element can be. In fact, the matrices you describe are a valid basis for the space of all $n\times n$ matrices. However, looking at matrices this way (as vectors of the vector space of all $n\times n$ matrices), it might help to realize that they are just tuples with $n^2$ many entries, arranged as a square.

Yes, you are right. A vector space of matrices of size $n$ is actually, a vector space of dimension $n^2$. In fact, just to spice things up: The vector space of all

- diagonal,
- symmetric and
- triangular matrices of dimension $n\times n$

is actually a subspace of the space of matrices of that size.

As with all subspaces, you can take any linear combination and stay within the space. (Also, null matrix is in all the above three).

Try to calculate the basis for the above 3 special cases: For the diagonal matrix, the basis is a set of $n$ matrices such that the $i^{th}$ basis matrix has $1$ in the $(i,i)$ and $0$ everywhere else.

Try to figure out the basis vectors/matrices for symmetric and triangular matrices.

- How to find integer solutions to $M^2=5N^2+2N+1$?
- Evaluate the sum $\sum_{k=0}^{\infty}\frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)}$?
- Fundamental group of a quotient on a solid torus.
- Show that $x^4 + 8$ is irreducible over Z
- Weak Convergence of Positive Part
- Limit of function at infinity: $ \lim_{n\to \infty} \frac{a^n-b^n}{a^n+b^n} $
- Finding the Fourier series of a piecewise function
- Find the value of $\lfloor x+y \rfloor$ where $x \in \mathbb{R}$, $y \in \mathbb{Z}$
- How to compute $\sum^n_{k=0}(-1)^k\binom{n}{k}k^n$
- seeing the differential dx/y on an elliptic curve as an element of the sheaf of differentials
- Partial fractions to integrate$\int \frac{4x^2 -20}{(2x+5)^3}dx$
- Does every number not ending with zero have a multiple without zero digits at all?
- Expected Value of Maximum of Two Lognormal Random Variables with One Source of Randomness
- Weak limit and strong limit
- Let $G$ a group with normal subgroups $M,N$ such that $M\cap N=\{e\}$. Show that if $G$ is generated by $M\cup N$ then $G\cong M \times N$.