MATH 110 - Day 13

2024-10-07 11:46:55 -0400 EDT

Subspaces

Recall: $R^n = { \vec{x} = ( x_1, … x_n ) | x_1, … x_n \in \mathbb{R}}$

Let $S$ be a subset of $\mathbb{R}^n$, we say that $S$ is a subspace (subvectorspace) of $\mathbb{R}^n$ is $S$ has the following properties:

  1. $ \vec{0} = (0, …, 0) \in S$
  2. if $\vec{x}, \vec{y} \in S, then \vec{x} + \vec{y} \in S.
  3. if $\vec{x} \in S$, and $c \in \mathbb{R}$, then $ c \vec{x} \in S$ (S is closed under scaler multiplication)

Example:

  1. $\phi$ (the empty set) NOT a subspace (violates 1.)
  2. Let $\mathbb{Z}^2 = \lbrace (x_1, x_2) | x_1, x_2 \in \mathbb{R} \rbrace$
    NOT a subspace (violates 3.)
  3. Let $S = \lbrace (x_1, x_2, x_3) \in \mathbb{R}^3 | x_3 = x_1^2 + x_2^2 \rbrace$
    NOT a subspace (violates 3.)
  4. Let $S = \lbrace (x_1, x_2, x_3) \in \mathbb{R}^3 | x_3 = 1 \rbrace$
    NOT a subspace (violates 1.)
  5. Let $S_1 = \lbrace (x, 0) \in \mathbb{R}^2 | x \in \mathbb{R} \rbrace$ or the X axis,
    $S_2 = \lbrace (0, y) \in \mathbb{R}^2 | y \in \mathbb{R} \rbrace$ or the Y axis.
    Both ARE Subspaces, but $S_1 \cup S_2$ is NOT

Rules about Subspaces

  1. $\lbrace \vec{0} \rbrace$ is a subspace of $\mathbb{R}^n$ called the trivial subspace
  2. Let $S_1, S_2$ be a subspace of $\mathbb{R}^n$, Let $\vec{u} \in S_1 \cap S_2 = \lbrace \vec{u} \in S_1 \text{ and } \vec{u} \in S_2 \rbrace$
  3. If $S_1, S_2, S_3, … S_k$ are subspaces of $\mathbb{R}^n$, then $S_1 \cap S_2 \cap S_3 \cap … \cap S_k \in \mathbb{R}^n$
  4. Let $\vec{u}_1, \vec{u}_2, \vec{u}_3, … \vec{u}_k$ be vectors in $\mathbb{R}^n$, then $span(\vec{u}_1, … \vec{u}_k) = \lbrace c_1 \vec{u}_1 + c_2 \vec{u}_2 + … + c_k \vec{u}_k \rbrace$
  5. Let $\delta := \lbrace (x, y, z) \in \mathbb{R}^n | ax + by + cz = d \rbrace$, then $\delta$ is a plane on $\mathbb{R}^3$ with a normal vector $(a, b, c)$. if $d = 0$, then $\delta$ is a subspace of $\mathbb{R}^3$, but if $d \neq 0$, $\delta$ is not subspace of $\mathbb{R}^3$.

Subspaces associated with Matrix

Let $m \times n$ matrix where $R_i$ is the $i$th row of $A$ and $R_i \in \mathbb{R}^n$. $C_j$ is the $j$th column of $A$ and $C_j \in \mathbb{R}^n$

Definition: The Row space of $A$, denoted by $Row(A)$, is a subspace of $\mathbb{R}^n$ spanned by rows of $A$ (i.e. $Row(A) = span( R_1, R_2, … R_m) \supseteq \mathbb{R}^n$)

Column space has the same definition but the values are replaced in context to the column: $ col(A) = span(C_1, C_2, … C_n) \in \mathbb{R}^m$

Theorem: Let $A$ be the $m \times n$ matrix, Let $E$ be the RREF of $A$, then $row(A) = row(E)$ (i.e. $row(A) \subseteq row(E)$ and $row(E) \subseteq row(A)$)

HOWEVER! $col(A) \neq col(E)$ (since elementary operations on done on rows and not columns)