Dimension and basis

How many $k$-dimensional subspaces, $1 \leqslant k \leqslant n$, does an $n$-dimensional vector space $V$ over a field $\mathbb {F}_{q}$ of $q$ elements have?

Determine the dimension of the space of real square matrices of order $n$:

What is the dimension of the space of all polynomials $f(t)$ of degree $\leqslant n$ in one variable satisfying $f(1)=0$? Find a basis of this space.

Prove that the sum $U=U_{1}+U_{2}+\ldots +U_{m}$ is direct precisely when

Find the transition matrix from the basis $\left(1, t, \ldots , t^{n-1}\right)$ of the space $P_{n}$ to the basis $\left(1,(t-\alpha ), \ldots ,(t-\alpha )^{n-1}\right)$ of the same space.

Let $\theta$ be a complex root of a polynomial $f \in \mathbb {Q}{}[t]$ irreducible over $\mathbb {Q}$. Find the dimension over $\mathbb {Q}$ of the space $\mathbb {Q}{}[\alpha ]= \left\langle 1, \theta , \ldots , \theta^{k}, \ldots \right\rangle_{\mathbb {Q}}$.

Prove that the cancellation law does not hold for direct sums, i.e., from the equality of sums $U \oplus W_{1}=U \oplus W_{2}$ with the same summand $U$, it does not follow, in general, that $W_{1}=W_{2}$.

Determine whether the quotient space $\mathfrak {K}[t] / L$ is finite-dimensional, where:

Prove the following analogue of the Grassmann formula:

($U$ and $W$ are subspaces of finite codimension of a not necessarily finite-dimensional vector space $V$).

Following the terminology of Example 8 from § 1, let us single out the trivial semi-magic matrices

Recall (Example 8 from § 1): a matrix $A \in M_{n}(\mathbb {Q})$ is called semi-magic (or a semi-magic square) if the sums of the coefficients in each row and in each column of the matrix coincide:

If, moreover, $\operatorname {tr} A=\sigma (A):=\sum_{i=1}^{n} a_{i, n+1-i}$, then the matrix $A$ is called magic (or a magic square). The set $\operatorname {SMag}_{n}(\mathbb {Q})$ of semi-magic squares, as well as the set $\operatorname {Mag}_{n}(\mathbb {Q})$ of all magic squares of order $n$, are vector spaces over $\mathbb {Q}$, and $\operatorname {Mag}_{n}(\mathbb {Q}) \subset \operatorname {SMag}_{n}(\mathbb {Q}) \subset M_{n}(\mathbb {Q})$.

The question arises: what are the dimensions $\operatorname {dim} \operatorname {SMag}_{n}(\mathbb {Q})$ and $\operatorname {dim} \operatorname {Mag}_{n}(\mathbb {Q})$? Obviously, $\operatorname {SMag}_{2}(\mathbb {Q})=\langle E, D\rangle_{\mathbb {Q}}$. In this case $S=E+D$ is the unique magic matrix, up to a rational factor. For $n=3$ one can exhibit a less obvious magic matrix

Compute the dimensions indicated above for $n=3$ and $n=4$.

Prove the decomposition into a direct sum

Let $V_{1}, \ldots , V_{k}$ be subspaces of an $n$-dimensional vector space $V$. Prove that if $\operatorname {dim} V_{1}+\ldots +\operatorname {dim} V_{k}>n(k-1)$, then $\bigcap_{i=1}^{k} U_{i} \neq \mathbf{0}$ (a direct generalization of the statements following from formula (7)).

Formula (7) (Theorem 6 of § 2): if $U$ and $W$ are finite-dimensional subspaces of a vector space $V$, then

Since the dimension of the sum $U+W$ does not exceed the dimension of the ambient space $V$, it follows from (7) that the intersection $U \cap W$ is nontrivial whenever $\operatorname {dim} U+\operatorname {dim} W>\operatorname {dim} V$ (for example, two planes of three-dimensional space necessarily contain a common line).