$\dim (f^{-1}(K \cap \operatorname{im} f))= \dim (\ker f)+ \dim(K \cap \operatorname{im} f)$
97040
Question about an application of the Rank-Nullity theorem
65923
How to approach proving $f^{-1}(B\setminus C)=A\setminus f^{-1}(C)$?
63036
Proving that $\dim(\mathrm{span}({I_n,A,A^2,...})) \leq n$
138069
Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $
77361
How to prove $|H|[K:K \cap (g^{-1}Hg)]=|K|[H:H \cap (gKg^{-1})]$?
155066
Dimension of the quotient space $\frac{C_{0}}{M}.$
137983
Normalisation of $k[x,y]/(y^2-x^2(x-1))$
88278
Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective
52750
Prove that $f^\ast\omega=\det f \cdot \omega$
13606
Prove that $K$ is isomorphic to a subfield of the ring of $n\times n$ matrices over $F$.
48291
Proof that $\mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $
75863
how to prove $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$
87494
$\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2\operatorname{rank}(T^2)$ for finite-dimensional $V$
46265
Minimal polynomial of the operator $T:V\oplus W\to V\oplus W$
45469
If $a_k = \operatorname{rank}(S^k)$, deduce that $a_k - a_{k+1} \geq a_{k+1} - a_{k+2}$
125184
Why $\dim(\hom(V,W))=\dim(V) * \dim(W)$?
23071
Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A =\displaystyle{ \sum_{i=1}^n} f_0(x_1, \ldots, Ax_i,\ldots, x_m)$