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Section 2.5 Subspaces

Definition 2.5.1.

A subset \(\mathbb{M}\) of a linear space \(\mathbb{L}\) is a subspace of \(\mathbb{L}\) if \(\mathbb{M}\) is itself a linear space. This means:

\(\mathbb{M}\subseteq \mathbb{L}\)

\(\bullet\) \(\mathbb{M}\) is closed under addition: \(\mathbf{P}\text{,}\) \(\mathbf{Q} \in \mathbb{M} \Rightarrow \mathbf{P} + \mathbf{Q} \in \mathbb{M}\)

\(\bullet\) \(\mathbb{M}\) is closed under scalar multiplication: \(c \in \mathbb{R}\text{,}\) \(\mathbf{P} \in \mathbb{M} \Rightarrow\) c\(\mathbf{P} \in \mathbb{M}\)

\(\mathbb{M}\) satisfies Axioms \(1- 7\) for linear spaces

Is \(\mathbb{R}^3 \subseteq \mathbb{R}^3\) ? Is \(\{\} \subseteq \mathbb{P}\) ? Is \(\mathbb{R}^2 \subseteq \mathbb{R}^3\) ? Is \(\mathbb{P} \subseteq\) C[0,1] ?

Write out several elements in the sets described using set notation below:

  1. \(\displaystyle \mathbb{S}=\{ x \; |\; 3x \in \mathbb{Z} \}\)

  2. \(\displaystyle \mathbb{S}=\{ 3x \; |\; x \in \mathbb{Z} \}\)

Which of the following sets \(\mathbb{M}\) are subspaces of the given linear space? If a set is not a subspace which properties of subspace does it fail to satisfy?

  1. \(\displaystyle \mathbb{M}=\{ \ot{x}{y}{0} \; |\; x,y \in \mathbb{R} \}\subseteq \mathbb{R}^3\)

  2. \(\displaystyle \mathbb{M}=\{ \ot{x}{y}{z} \;|\; z=x+y,\;x,y\in \mathbb{R}\}=\{\ot{x}{y}{x+y}\;|\; x,y \in \mathbb{R}\}\subseteq \mathbb{R}^3\)

  3. \(\displaystyle \mathbb{M}=\{ \ot{x}{y}{z} \; |\; x,y,z \in \mathbb{R}^+ \}\subseteq \mathbb{R}^3\)

  4. \(\displaystyle \mathbb{M}=\{ c_0+c_1x+c_2x^2\; |\; c_i\in \mathbb{Z} \}\subseteq \mathbb{P}\)

  5. \(\displaystyle \mathbb{M}=\{ f(x)\in C[0,1]\; |\; f(0) = \frac{1}{4} \}\subseteq C[0,1]\)

  6. \(\displaystyle \mathbb{M}=\{ f(x)\in C[0,1]\; |\; f(\frac{1}{4}) = 0 \}\subseteq C[0,1]\)

  7. \(\displaystyle \mathbb{M}=\{ f(x)\in C[0,1]\; |\; f\ '(x) \in C[0,1] \}\subseteq C[0,1]\)