Section01 Spaces of Vectors

Spaces

• Defination The space $\mathbb{R}^{n}$ consists of all column vectors $\boldsymbol{v}$ with $n$ components.
  • The components of $\boldsymbol{v}$ are real numbers.
  • A vector whose $n$ components are complex numbers lies in the $\mathbb{C}^{n}$

• Perspectives of Space
  • We can add any vectors in $\mathbb{R}^{n}$, and we can multiply any vector $\boldsymbol{v}$ by any scalar $c$
  • The result stays in the spaces

• Eight rules of vector addition and scalar multiplication.
  1. $\boldsymbol{x} + \boldsymbol{y} = \boldsymbol{y} + \boldsymbol{x}$
  2. $\boldsymbol{x} + (\boldsymbol{y} + \boldsymbol{z}) = (\boldsymbol{x} + \boldsymbol{y}) + \boldsymbol{z}$
  3. $\text{There is a unique "zero vector" such that } \boldsymbol{x}+ \boldsymbol{0} = \boldsymbol{x} \text{ for all } \boldsymbol{x}.$
  4. $\text{For each } \boldsymbol{x} \text{ there is a unique vector } -\boldsymbol{x} \text{ such that } \boldsymbol{x} + (-\boldsymbol{x}) = \boldsymbol{ 0 }$
  5. $1 \text{ times } \boldsymbol{x} \text{ equals } \boldsymbol{x}$
  6. $(c_{1}c_{2})\boldsymbol{x} = c_{1}(c_{2}\boldsymbol{x})$
  7. $c(\boldsymbol{x} + \boldsymbol{y}) = c \boldsymbol{x} + c \boldsymbol{y}$
  8. $(c_{1} + c_{2}) \boldsymbol{x} = c_{1} \boldsymbol{x} + c_{2} \boldsymbol{x}$

Subspaces

• A vector space inside $\mathbb{R}^{n}$ (it isn't $\mathbb{R}^{n-1}$, because the vectors of it have $n$ components.)
• Defination
  • A Subspace of a vector is a set of vectors (including $\boldsymbol{0}$) that statisfies two requirements: If $\boldsymbol{v}$ and $\boldsymbol{w}$ are vectors in the subspace and $\boldsymbol{c}$ is any scalar, then
    1. $\boldsymbol{v}+ \boldsymbol{w}$ is in the subspace.
    2. $c\boldsymbol{v}$ is in the subspace.
  • A subspace containing $\boldsymbol{v}$ and $\boldsymbol{w}$ must contain all linear combination of $c \boldsymbol{v} + d \boldsymbol{w}$
• Facts
  1. Every subspaces contains the zero vector.
  2. Space is a subspace of itself.
  3. Quarter-plane is not a subspace. (Because $-\boldsymbol{v}$ is not lie in the subspace.)

• The whole subspaces of $\mathbb{R}^{2}$

  1. ($\mathbf{L}$) Any line through $(0,0,0)$.
  2. ($\mathbb{R}^{3}$) The whole space.
  3. ($\mathbf{P}$) Any plane through $(0,0,0)$.
  4. ($\mathbf{Z}$) The single vector $(0,0,0)$.

• Subspaces of vector space $\boldsymbol{M} (2 \times 2)$

  • ($\boldsymbol{U}$) $\left[ \begin{matrix} a & b \\ 0 & c \end{matrix} \right]$
  • ($\boldsymbol{D}$) $\left[ \begin{matrix} a & 0 \\ 0 & c \end{matrix} \right]$
  • ($\boldsymbol{Z}$) $\left[ \begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix} \right]$

The Column Space of $\boldsymbol{A}$

• Defination
  • The Column space consists of all linear combinations of the columns. The combination are all possible vectors $\boldsymbol{Ax}$. Thy fill the column space $\boldsymbol{C}(\boldsymbol{A})$
• The system $\boldsymbol{Ax} = \boldsymbol{b}$ is solvable if and only if $\boldsymbol{b}$ is in the column space of $\boldsymbol{A}$
• Suppose $\boldsymbol{A}$ is a $m \times n$ matrix. Its column has $m$ components. The column space of $\boldsymbol{A}$ is the subspaces of $\mathbb{R}^{2}$.
• $\boldsymbol{S}$ and $\boldsymbol{SS}$
  • $\boldsymbol{S}$: set of vectors in $\boldsymbol{V} (\big\{ \boldsymbol{a}_{\boldsymbol{\cdot}1}, \boldsymbol{a}_{\boldsymbol{\cdot}2}, \cdots \boldsymbol{a}_{\boldsymbol{\cdot}n} \big\} )$
  • $\boldsymbol{SS}$: all combinations of vectors in $\boldsymbol{S}$
  • The subspace $\boldsymbol{SS}$ is the "span" of $\boldsymbol{S}$, containing all combinations of vectos in $\boldsymbol{S}$.

Properties from Exercise

• $\boldsymbol{V} \text{ and } \boldsymbol{S} \text{ and } \boldsymbol{SS}$

  • $\boldsymbol{V} = \text{ all solution to equation } d^{4}/dx^{4} = 0$
    • $\boldsymbol{V}$ contains all cubic polynomials $y = a + bx + cx^{2} + dx^{3} \text{ with } d^{4}y/dx^{4} = 0$, therefore this cubic polynomials is a subspace $\boldsymbol{S}$ of $\boldsymbol{V}$.
    • The linear polynomials are the subspace of $\boldsymbol{S}$ ($\boldsymbol{SS}$).
    • The constant could be the subspace of linear polynomials. ($\boldsymbol{SSS}$)

• symmetric and skew-symmetric

  • The symmetric matrices in $\boldsymbol{M}$ form a subspaces.(The linear combination of symmetric matrices is also symmetric.)
  • The skew-symmetric matrices in $\boldsymbol{M}(\boldsymbol{A}^{T} = -\boldsymbol{A})$ form a subspaces. (The linear combination of skew-symmetric matrices is also skew-symmetric.)

• Linear Combination of Rows and Columns

  • Rows of $\boldsymbol{B}$ is the linear combination of rows of $\boldsymbol{A}$.
    • $\boldsymbol{A}$ and $\boldsymbol{B}$ may not have the same column space.
  • Columns of $\boldsymbol{C}$ is the linear cobination of columns of $\boldsymbol{A}$.
    • $\boldsymbol{A}$ and $\boldsymbol{C}$ have the same column space. (the linear combination of $\boldsymbol{C}$'s columns is also the linear combination of $\boldsymbol{A}$'s columns.)

• Add an extra column $\boldsymbol{b}$ to matrix $\boldsymbol{A}$ (augment matrix $\left[ \begin{matrix} \boldsymbol{A} & \boldsymbol{b} \end{matrix} \right]$)

  • when $\boldsymbol{b}$ is the linear combination of $\boldsymbol{A}$'s columns, the augment matrix's column space will not be larger than $\boldsymbol{A}$
  • It's solvable when the augmnet matix $\left[ \begin{matrix} \boldsymbol{A} & \boldsymbol{b} \end{matrix} \right]$'s columns space isn't larger than the origin matrix $\boldsymbol{A}$'s.

• The column space of $\boldsymbol{AB}$ is contained in the column space of $\boldsymbol{A}$

• $\boldsymbol{S} \text{ and } \boldsymbol{T}$ are two subspaces of a vector space $\boldsymbol{V}$

  • all sums $\boldsymbol{s} + \boldsymbol{t}$ of a vector $\boldsymbol{s}$ ins $\boldsymbol{S}$ and a vector $\boldsymbol{t}$ in $\boldsymbol{T}$ is contained in $\boldsymbol{S} + \boldsymbol{T}$
  • The difference between $+$ and $\cup$
    • Suppose $\boldsymbol{S}$ and $\boldsymbol{T}$ are lines in $\mathbb{R}^{m}$
    • $\boldsymbol{S} \cup \boldsymbol{T}$ is just two lines.
    • $\boldsymbol{S} + \boldsymbol{T}$ is the whole plane that they span.

• If $\boldsymbol{S} = \boldsymbol{C}(\boldsymbol{A}); \boldsymbol{T} = \boldsymbol{C}(\boldsymbol{B})$ then $\boldsymbol{S} + \boldsymbol{T} = \boldsymbol{C}(\left[ \begin{matrix} \boldsymbol{A} & \boldsymbol{B} \end{matrix} \right])$
  • When $\boldsymbol{S}$ is the column space of $\boldsymbol{A}$, $\boldsymbol{T}$ is the column space of $\boldsymbol{B}$, then $\boldsymbol{S} + \boldsymbol{T}$ is the column space of augment matrix $\left[ \begin{matrix} \boldsymbol{A} & \boldsymbol{B} \end{matrix} \right]$