Intereting Posts

Image of Matrix Exponential Map
Cardinality of set of real continuous functions
Why is the trace of a matrix the sum along its diagonal?
Find all solutions to $x^3+(x+1)^3+ \dots + (x+15)^3=y^3$
Serendipitous mathematical discoveries in recent times
The rule of three steps for a cyclically ordered group
Expression for power of a natural number in terms of binomial coefficients
Describing all holomorphic functions such that $f(n)=n$ for $n \in \mathbb{N}$
Properties of $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$ where $\{a_n\}$ is unbounded, strictly increasing sequence of positive reals
how to find the root of permutation
What went wrong?
Existence of fixed point given a mapping
Problem based on sum of reciprocal of $n^{th}$ roots of unity
Reduced frequency range FFT
Is $SL(2, 3) $ a subgroup of $SL(2, p)$ for $ p>3$?

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?

- Finding $\det(I+A^{100})$ where $A\in M_3(R)$ and eigenvalues of $A$ are $\{-1,0,1\}$
- Frobenius Inequality Rank
- Linear dependence of $\left\{x^{n}\,\colon\, n\in\mathbb{N}\right\}$
- Proving that two systems of linear equations are equivalent if they have the same solutions
- Exact solution of overdetermined linear system
- Determinant of a finite-dimensional matrix in terms of trace

- Fredholm Alternative as seen in PDEs, part 2
- How to diagonalize this matrix…
- Proving that every $2\times 2$ matrix $A$ with $A^2 = -I$ is similar to a given matrix
- Representing natural numbers as matrices by use of $\otimes$
- Writing up a rigorous solution for finding a basis for the $n \times n$ symmetric matrices.
- How do I prove that a subspace of a vector space $X$ is the null space of some linear functional on $X$?
- Determining matrix $A$ and $B$, rectangular matrix
- Basis for dual in infinite dimensional vector space.
- Every matrix can be written as a sum of unitary matrices?
- Deriving volume of parallelepiped as a function of edge lengths and angles between the edges

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.

- Convergence of the series $\sum_{n = – \infty}^{\infty} (\sqrt{(a+n)^2+b^2} – |n| )$
- What does orthogonality mean in function space?
- Proving Integral Inequality
- How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
- Is $\mathbb{R}/\mathord{\sim}$ Hausdorff if $\{(x,y)\!:x\sim y\}$ is a closed subset of $\mathbb{R}\times\mathbb{R}$?
- Is a function whose derivative vanishes at rationals constant?
- Find $\lim \limits_{x\to 0}\frac{\log\left(\cos x\right)}{x^2}$ without L'Hopital
- Find the eigenvalues and eigenvectors with zeroes on the diagonal and ones everywhere else.
- Are the axioms for abelian group theory independent?
- Showing $(a+b+c)(x+y+z)=ax+by+cz$ given other facts
- Numbers of circles around a circle
- Weird $3^n$ in an identity to be combinatorially proved
- Algebraic integers of a cubic extension
- Separately continuous functions that are discontinuous at every point
- The area of the region $|x-ay| \le c$ for $0 \le x \le 1$ and $0 \le y \le 1$