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Prove That...

C'mon, prove these linear algebra things!

For the content below, I use \(W\leqslant V\) to denote \(W\) is a subspace of \(V\), and \(\exists!\) to denote "there exists a unique".

Group, Ring & Field

  1. \(\mathbb{Q}(\sqrt{2}) = \{a+b\sqrt{2}\mid a,b\in\mathbb{Q}\}\) is a field.
  2. \(\mathbb{Q}\) is the smallest number field (i.e. subfield of \(\mathbb{C}\)).

Linear Space

  1. For \(W\subset V(\mathbf{F})\), \(W\leqslant V\iff W\) is closed under addition and scalar multiplication.
  2. \(S_1\subset S_2\subset V(\mathbf{F})\Rightarrow\operatorname{span}S_1\subset\operatorname{span}S_2\).
  3. If \(S_1\) and \(S_2\) are linearly independent, then \(\operatorname{span}S_1=\operatorname{span}S_2\Rightarrow|S_1|=|S_2|\).
  4. Functions \(\exp(\lambda_1x)\), \(\exp(\lambda_2x)\) and \(\exp(\lambda_3x)\) are linearly independent. (\(\lambda_1\), \(\lambda_2\) and \(\lambda_3\) are distinct)
  5. For \(W_1, W_2\leqslant V(\mathbf{F})\):
    1. \(W_1\cup W_2\leqslant V\iff W_1\subset W_2\) or \(W_2\subset W_1\).
    2. \(W_1 + W_2 = \operatorname{span}(W_1\cup W_2)\).
    3. \(\dim(W_1 + W_2) = \dim W_1 + \dim W_2 - \dim(W_1\cap W_2)\).
  6. For \(W_1, W_2\leqslant V(\mathbf{F})\), the following propositions are equivalent:
    1. \(W_1 \cap W_2 = \{0\}\).
    2. \(\forall\alpha\in W_1 + W_2\), \(\exists!\alpha_1\in W_1\) and \(\exists!\alpha_2\in W_2\) such that \(\alpha = \alpha_1 + \alpha_2\).
    3. If \(0 = \alpha_1 + \alpha_2 (\alpha_1 \in W_1, \alpha_2 \in W_2)\), then \(\alpha_1 = \alpha_2 = 0\).
    4. \(\dim(W_1 + W_2) = \dim W_1 + \dim W_2\).

Inner Product Space

  1. (Cauchy–Schwarz inequality) \(|\langle\alpha,\beta\rangle|\leq \|\alpha\|\cdot\|\beta\|\).
  2. (Triangle inequality) \(\|\alpha\|+\|\beta\| \geq \|\alpha+\beta\|\).
  3. (Pythagorean theorem) \(\|\alpha\|^2+\|\beta\|^2 = \|\alpha+\beta\|^2\iff\alpha\perp\beta\iff\angle(\alpha,\beta)=\dfrac{\pi}{2}\).
  4. (Gram–Schmidt process) Any Euclidean space has an orthonormal basis. (The method to construct it is called Gram–Schmidt process.)
  5. If \(B = \{\varepsilon_1, \varepsilon_2, \cdots, \varepsilon_n\}\) is an orthonormal basis of \(V(\mathbf{F})\), then \(\forall\alpha\in V\), \(\alpha = \sum\limits_{i=1}^n\langle\alpha,\varepsilon_i\rangle\cdot\varepsilon_i\).

Linear Transformation

  1. \(\sigma:V\to W\) is injective \(\iff\ker\sigma = \{0\}\).
  2. \(\sigma:V\to W\) is surjective \(\iff\operatorname{im}\sigma = W\).
  3. If \(B=\{\alpha_1,\alpha_2,\dots,\alpha_n\}\) is a basis of \(V\), then \(\forall \beta_1, \beta_2, \dots, \beta_n\in W\), \(\exists !\sigma:V \to W\) such that \(\sigma(\alpha_i) = \beta_i\).
  4. For \(\sigma: V \to W\), \(\operatorname{rank}\sigma + \dim\ker\sigma = \dim V\).
  5. For \(\sigma: V \to W\), if \(\dim V = \dim W = n\), the following propositions are equivalent:
    1. \(\sigma\) is injective.
    2. \(\sigma\) is surjective.
    3. \(\operatorname{rank}\sigma = n\).
  6. \(V\cong W \iff \dim V = \dim W\).

Matrix

  1. \(\mathbf{M}(\sigma)\) and \(\sigma\) are one-to-one.
  2. Prove with matrix: if \(\dim V(\mathbf{F})=m\), \(\dim W(\mathbf{F})=n\), then \(\mathcal{L}(V, W)\cong \mathbf{F}^{m\times n}\).
  3. Prove with matrix: \(\dim\mathcal{L}(V, W)=\dim V\cdot\dim W\).
Last update: 2023-11-13
Created: 2023-11-07