I understood basis as a set of vector $v_{1},v_{2},...,v_{n}$ as the set whose linear combination will span the entire vector space say $ \mathbb R^{n}$ which makes perfect sense in intuitive terms. There are $n$ independent vectors, linear combination spans the entire space and the dimensions equals the number of linearly independent vector.
But I came across a very interesting question, which asked, is this a vector space? If yes, then find its dimension and basis. and asks this about,
- all skew symmetric matrices of $2 \times 2$ dimension
Its interesting to note, it is associative,commutative under addition operator and scaling of the same is a subset of the same space. (do correct me if my choice of words is right here)
Which implies its a vector space, with a basis of $\begin{bmatrix} 0 & 1\\ -1 &0 \end{bmatrix}$ That implies dimension is 1.
So my question is, if basis indeed can span a linear system (represented by the matrix)? If my interpretation is right, then can anyone give me an intuitive "feel" of the basis, dimension and vector space.
Help much appreciated