Prove that $ \dim(\ker(g \circ f)) = \dim(\ker(f)) + \dim(\ker(g) \cap \operatorname{im}(f)). $ What is the error in this approach?
118025
Kernel of composition of linear transformations
132653
How to show that $\mathfrak{sl}_n(\mathbb{R})$ and $\mathfrak{sl}_n(\mathbb{C})$ are simple?
65162
Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ otherwise.
53364
dim$(V)$ = $n$, dim$(W)$ = $m$ $\implies$ dim($L(V,W)$) = $nm$
150720
Given that $W$ is a subspace of $V$, prove: $\dim W=\dim V \implies W=V$
80892
Does $\wp(A \cap B) = \wp(A) \cap \wp(B)$ hold? How to prove it?
155066
Dimension of the quotient space $\frac{C_{0}}{M}.$
103380
Is $\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$ true?
130537
Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$
81181
Prove that $\det(M-I)=0$ if $\det(M)=1$ and $MM^T=I$
1162
Why $\{\mathbf{0}\}$ has dimension zero?
22758
If $\operatorname{ran} F \subseteq \operatorname{dom} G$, then $\operatorname{dom}(G \circ F) = \operatorname{dom} F$.
63036
Proving that $\dim(\mathrm{span}({I_n,A,A^2,...})) \leq n$
117152
Preimage of codimension one subspace of the quotient $g/D^1 g$
120985
Prove that $\mathrm{Res}[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$
65395
Prove that if $\operatorname{rank}(T) = \operatorname{rank}(T^2)$ then $R(T) \cap N(T) = \{0\}$
11557
How to prove $\det(e^A) = e^{\operatorname{tr}(A)}$?