I do not know if this is the right place to do it but I am looking for theoretical Linear Algebra I exercises. Of course they can admit some calculations but more focused on theory, could be even better if they contain 6-7 questions to maybe show a theorem or whatever (I will give a couple of examples at the end). The subjects are classic Linear Algebra 1 ones : groups, rings, fields, vector spaces, linear applications, matrixes (especially companion matrix), complex numbers, row echelon form, multilinear forms and determinants.
Example 1 : Let $\mathbb Z [i] := \mathbb Z + \mathbb Z.i = \{a + ib, \text{with } a, b ∈ \mathbb Z\} ⊂ \mathbb C$ (the Gaussian integers).
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- Show that $\mathbb Z [i]$ is a subring of $\mathbb C$.
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- Show that $\mathbb Z [i]^{\times} = \{a+ib \in \mathbb Z [i], \lvert a +ib \rvert=1\} $.
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- Show that $\mathbb Z [i]^{\times} = \{±1,±i\}$.
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- Let $w = u + iv \in \mathbb C$ and the square $C_w := [u − 1/2, u + 1/2] + i[v − 1/2, v + 1/2] ⊂ \mathbb C$. Show that $C_w$ contains at least one Gaussian integer.
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- Let $q \in \mathbb Z [i] - \{0\}$ a nonzero Gaussian integer and $z \in \mathbb Z [i]$ another Gaussian integer. Show that $\exists$ Gaussian integers $k,r$ such that $z = q.k + r$ and $|r| < |q|$.
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- Let $I \neq \{0\}$ and $I ⊂ \mathbb Z [i]$ be an ideal of $\mathbb Z [i]$. Show that $\exists q \in I −\{0\}$ of minimal module $|q|$ among all nonzero elements of I.
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- Show that $I = q.\mathbb Z [i] = \{q.k, k \in \mathbb Z [i]\}$.
Example 2 : Let $\phi : U \to V$ , $\psi : V \to W$ linear applications between finite vector spaces.
- Show that $Im(\psi \circ \phi) ⊂ Im(\psi)$.
- Show that $ker(\phi) ⊂ ker(\psi \circ \phi)$.
- Show that $rank(\psi \circ \phi) \leq min(rank(\psi),rank(\phi))$.
- We suppose that $U = V = W$. Show that $rank(\psi \circ \phi) \geq rank(\psi) + rank(\phi) − dim(V)$.