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Denote by $E_{i j}$ the $m \times n$ matrix whose entry at the intersection of the $i$-th row and the $j$-th column equals 1 , and all the remaining entries are zero. Verify that these $mn$ matrices form a basis in the linear space of $m \times n$ matrices. (This basis is called the standard basis of the given space.) What are the coordinates of a matrix $A$ with entries $a_{i j}$ in the standard basis?

Prove that the upper triangular matrices of order $n$ form a linear space with respect to the usual operations on matrices. Find the dimension of this space and some basis in it.

In the linear space of polynomials of degree $\leqslant 3$ in the variable $t$ two bases are given: $1, t, t^{2}, t^{3}$ and $1, t-a,(t-a)^{2}$, $(t-a)^{3}$. Find the change-of-basis matrix from the first basis to the second and, using it, the expansion of a polynomial $p(t)$ in the second basis.

How are two bases $e_{1}, \ldots , e_{n}$ and $f_{1}, \ldots , f_{n}$ positioned relative to each other if the change-of-basis matrix from $\mathbf{e}$ to $\mathbf{f}$ is upper triangular? Prove from these considerations that the inverse of an upper triangular matrix is also upper triangular.

How are two bases oriented relative to each other if $f_{1}=e_{1}+e_{2} ; f_{2}=e_{2}+e_{3} ; f_{3}=e_{3}+e_{4} ; f_{4}=e_{4}-e_{1}$ ?