12.13 Change basis

1. Suppose \(V\) and \(W\) are finite-dimensional. Let \(T \in \mathcal{L}(V, W)\) and suppose there exists \(\varphi \in V^*\) such that \(\text{Range}(T^*) = \langle \varphi \rangle\). Prove that \(\text{Null}(T) = \text{Null}(\varphi)\).

   Since we know \(\text{Range}T^{*}=(\text{Null}T)^{\circ}\), then \((\text{Null}T)^{\circ}=\langle\varphi\rangle\), by definition: \(\{\varphi\in V^{*}:\varphi(v)=0,\forall v\in\text{Null}T\}=\langle\varphi\rangle\)​

   By previous tutorial exercise \(\text{Null}(T)=\{v\in V:\varphi(v)=0\;\forall\varphi\in\text{Null}T^{\circ}\}=\left\lbrace v\in V:\alpha\varphi(v)=0\;\right\rbrace=\left\lbrace v\in V:\varphi(v)=0\;\right\rbrace=\text{Null}(\varphi)\)

2. Suppose \(U\) is a subspace of \(V\). Let \(\pi : V \to V / U\) be the usual quotient map. Thus \(\pi^* \in \mathcal{L}((V/U)^*, V^*)\).

   1. Show that \(\pi^*\) is injective.

      Take \(\pi^*(\varphi_1)=\pi^*(\varphi_2)\), then \(\varphi_1\circ\pi=\varphi_2\circ\pi\)​

      Then consider any \(v\in V\) such that \(\varphi_1\left(\pi\left(v\right)\right)=\varphi_2\left(\pi\left(v\right)\right)\Rightarrow\varphi_1\left(\left\lbrack v\right\rbrack\right)=\varphi_2\left(\left\lbrack v\right\rbrack\right)\Rightarrow\varphi_1=\varphi_2\)​ 2. Show that \(\text{Range}(\pi^*) = U^0\).

      Since \(\text{Range}\pi^{*}=(\text{Null}\pi)^{\circ}\), then N.T.P. \(\text{Null}\pi=U\)​

      \(\subseteq\)) Take any \(v\in\text{Null}\pi\), then \(\pi(v)=0\Rightarrow\left\lbrack v\right\rbrack=0\Rightarrow v+U=0\Rightarrow v\in U\)​

      \(\supseteq\)) Take any \(v\in U\Rightarrow v\in\left\lbrack0\right\rbrack\Rightarrow v\in\text{Null}\pi\)​

      Thus \(\text{Range}(\pi^*) = U^0\)​ 3. Conclude that \(\pi^*\) is an isomorphism from \((V/U)^*\) onto \(U^0\).

      Yes

3. Suppose \(U, W\) are subspaces of \(V\).

   1. Show that \((U + W)^0 = U^0 \cap W^0\).

      \(\subseteq\)) Take \(\varphi\in V^*\) such that \(\varphi(u+w)=0,\forall u+w\in U+W\). Then if \(u=0\), we have \(\varphi \in U^\circ\).

      If \(w=0\), then we have \(\varphi\in W^\circ\). Thus \(\varphi \in U^\circ\) and \(\varphi\in W^\circ\)​

      \(\supseteq\)) Take \(\varphi\) in \(U^\circ\) and \(W^\circ\), then we have \(\varphi\in V^*\) such that \(\varphi(u)=0,\forall u\in U\) and \(\varphi(w)=0,\forall w\in U\)​

      Thus \(\forall v=u+w\in U+W\), we have \(\varphi(v)=\varphi(u)+\varphi(w)=0\)​ 2. Suppose \(V\) is finite-dimensional. Prove that \((U \cap W)^0 = U^0 + W^0\).

      Since \(\dim(U\cap W)^0=\dim V-\dim U\cap W\) and

      \(\dim\left(U^0+W^0\right)=\dim U^0+\dim W^0-\dim U^0\cap W^0=\dim U^0+\dim W^0-\dim(U+W)^0\)​

      \(=\dim V-\dim U+\dim V-\dim W-\dim V+\dim\left(U+W\right)=\dim V-\dim U-\dim W+\dim U+\dim W-\dim U\cap W\)​

      \(=\dim V-\dim U\cap W\)​

      Thus they are equal

4. Let \(\mathcal{B} = \left\{ \begin{pmatrix} 0 & 2 \\ -3 & 3 \end{pmatrix}, \begin{pmatrix} 0 & -1 \\ 2 & 0 \end{pmatrix}, \begin{pmatrix} 2 & 1 \\ -2 & 1 \end{pmatrix}, \begin{pmatrix} 4 & -2 \\ 2 & -3 \end{pmatrix} \right\}\).

   1. Prove that \(\mathcal{B}\) is a basis of \(M_2(\mathbb{R})\).
      Consider \(\begin{bmatrix}x & y \\z & w\end{bmatrix}= a \begin{bmatrix}0 & 2 \\-3 & 3\end{bmatrix}+ b \begin{bmatrix}0 & -1 \\2 & 0\end{bmatrix}+ c \begin{bmatrix}2 & 1 \\-2 & 1\end{bmatrix}+ d \begin{bmatrix}4 & -2 \\2 & -3\end{bmatrix}\)

      \(a = -x - 10y - 5z + 2w\) \(b=\frac{x}{2}+3y+2z\) \(c = \frac{3}{2}x + 12y + 6z - 2w\) \(d = -\frac{x}{2} - 6y - 3z + w\)​ 2. Find the change of basis matrix from the canonical basis \(C\) to \(\mathcal{B}\) and the change of basis matrix from \(\mathcal{B}\) to \(C\). 3. Find the coordinates of any \(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in M_2(\mathbb{R})\) in \(\mathcal{B}\).

5. Suppose \(T \in \mathcal{L}(V, V)\) and \(\mathcal{B} = \{v_1, \ldots, v_n\}\) and \(\mathcal{B}' = \{u_1, \ldots, u_n\}\) are bases of \(V\). Prove that the following are equivalent:

   1. \(T\) is invertible.
   2. The columns of \([T]_{\mathcal{B}, \mathcal{B}'}\) are linearly independent in \(\mathbb{F}^n\).
   3. The columns of \([T]_{\mathcal{B}, \mathcal{B}'}\) span \(\mathbb{F}^n\).
   4. The rows of \([T]_{\mathcal{B}, \mathcal{B}'}\) are linearly independent in \(\mathbb{F}^n\).
   5. The rows of \([T]_{\mathcal{B}, \mathcal{B}'}\) span \(M_{1 \times n}(\mathbb{F})\).

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