ALG1_T2

Properties

Let $V_1$ and $V_2$ be subspaces of $V$.

• Intersection of subspaces is a subspace.
• Union os subspaces is not a subspace.
• Lemma: $V_1\cup V_2\text{ is a subspace }⟺v_1\subset V_2\text{ or }{V}_2\subset V_1$.
• $V_1+V_2:=\{{\mathbf{v}}_1+{\mathbf{v}}_2|{\mathbf{v}}_1\in V_1,{\mathbf{v}}_2\in V_2\}$
• If $V_1\cap V_2=\{0\}$, and let $V^{\prime }=V_1+V_2$, (this is called a direct sum, when the intersection of $V_1$ and $V_2$ is $\{0\}$) then ${\mathbf{v}}_1(\ne 0)\in V_1$ and ${\mathbf{v}}_2(\ne 0)\in V_2$ are a basis of $V^{\prime }$.
• Let $T:V_1\times V_2\to V^{\prime }$ be a linear map, with addition and multiplication by a scalar on the space $V_1\times V_2$ being defined slot-wise. $T$ maps $({\mathbf{v}}_1,{\mathbf{v}}_2)$ to ${\mathbf{v}}_1+{\mathbf{v}}_2$. Show that $T$ is an isomorphism.
• Let $P$ be a linear map from $V$ to $V$. Let $P^2=P$. Every map of this form is a “projection”. Show that $V=\text{Ker }P+\mathrm{Im}P$.
  Show that (or is it given? idk) $P({\mathbf{v}}_1+{\mathbf{v}}_2)={\mathbf{v}}_2$, where ${\mathbf{v}}_1$ is in the kernel of $P$ and ${\mathbf{v}}_2$ is in the Image of $P$.
  Show that $\mathbf{v}-P\mathbf{v}\in \text{Ker }P$.