Vectors in \( \mathbb{R}^n \)

\[ \vec{u} = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}, \qquad \vec{v} = \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix} \qquad \in \mathbb{R}^n \]

The symbol \(\in\) means belong to.

Vector Addition

\[ \vec{u} + \vec{v} = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} + \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix} = \begin{pmatrix} x_1 + y_1 \\ x_2 + y_2 \\ \vdots \\ x_n + y_n \end{pmatrix} \]
Vectors u and v from the origin with u plus v drawn as the diagonal of a parallelogram.
Figure 1
A hand-drawn sketch on blue coordinate axes with the origin at lower left. Two black arrows start at the origin: one steep, nearly vertical arrow labeled \( \vec{v} \), and one shallow arrow pointing right and slightly upward labeled \( \vec{u} \). A red arrow from the origin runs diagonally up and to the right, ending at the far corner; it is labeled \( \vec{u} + \vec{v} \). A red dashed segment connects the tip of \( \vec{v} \) to that same far corner, and a second red arrow labeled \( \vec{v} \) rises from the tip of \( \vec{u} \) to the far corner, completing the parallelogram formed by \( \vec{u} \) and \( \vec{v} \). The figure illustrates the parallelogram or triangular law for vector addition.

parallelogram, or triangular law

Scalar Multiplication

\[ \begin{gathered} \alpha \in \mathbb{R}, \\[0.5em] \alpha \vec{u} = \alpha \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} = \begin{pmatrix} \alpha x_1 \\ \alpha x_2 \\ \vdots \\ \alpha x_n \end{pmatrix} \end{gathered} \]
Axes with vector u and the line through it showing alpha u for positive and negative alpha.
Figure 2
A hand-drawn coordinate diagram with blue horizontal and vertical axes crossing at the origin. A short black arrow labeled \( \vec{u} \) points from the origin up and to the right into the first quadrant. A red line passes through the origin along the same direction as \( \vec{u} \), extending as an arrow farther up and to the right into the first quadrant, labeled \( \alpha \vec{u} \) with the condition \( (\alpha > 0) \), and extending as an arrow down and to the left into the third quadrant, labeled \( \alpha \vec{u} \) with the condition \( (\alpha < 0) \). The diagram shows that scalar multiplication keeps the vector on the same line through the origin, pointing the same way for positive scalars and the opposite way for negative scalars.

Properties of Vector Add. & Scalar Mult.

\[ ( \vec{u} + \vec{v} \in \mathbb{R}^n, \qquad \alpha \vec{u} \in \mathbb{R}^n ) \]
  1. \[ \vec{u} + \vec{v} = \vec{v} + \vec{u} \]
  2. \[ (\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w}) \]
  3. \[ \alpha (\beta \vec{u}) = (\alpha \beta) \vec{u} \]
  4. \[ \alpha (\vec{u} + \vec{v}) = \alpha \vec{u} + \alpha \vec{v} \]
  5. \[ (\alpha + \beta) \vec{u} = \alpha \vec{u} + \alpha \vec{v} \]

  6. Let

    \[ \vec{0} = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{pmatrix} \]

    Then \( \vec{0} + \vec{u} = \vec{u} \)

  7. Let

    \[ -\vec{u} = \begin{pmatrix} -x_1 \\ -x_2 \\ \vdots \\ -x_n \end{pmatrix} \]

    Then \( (-\vec{u}) + \vec{u} = \vec{0} \)

  8. \[ 1 \vec{u} = \vec{u} \]

Linear Combination and Span

(1) Given \( \mathscr{L} = \{ \vec{u}_1, \vec{u}_2, \ldots, \vec{u}_k \} \subseteq \mathbb{R}^n \). \( \vec{v} \in \mathbb{R}^n \) is said to be a linear combination of \( \mathscr{L} \) if there are scalars \( c_1, c_2, \ldots, c_k \) such that

\[ \vec{v} = c_1 \vec{u}_1 + c_2 \vec{u}_2 + \cdots + c_k \vec{u}_k \]
Chain of vectors c1 u1, c2 u2, up to ck uk joined tip to tail, with red vector v from start to end.
Figure 3
A hand-drawn diagram illustrating a linear combination as a tip-to-tail chain of scaled vectors. Starting from a point at the lower left, a short black arrow labeled \( c_1 \vec{u}_1 \) points to the right. From its tip, another black arrow labeled \( c_2 \vec{u}_2 \) rises up and to the right, then descends, followed by additional zig-zagging black arrows representing intermediate terms, and finally an arrow labeled \( c_k \vec{u}_k \) at the upper right. A long straight red arrow drawn from the original starting point directly to the final tip is labeled \( \vec{v} \), showing that the sum of the scaled vectors equals \( \vec{v} \).

Linear Combination and Span

(2) Given \( \mathscr{L} = \{ \vec{u}_1, \vec{u}_2, \ldots, \vec{u}_k \} \subseteq \mathbb{R}^n \).

\[ \mathrm{Span}\{ \vec{u}_1, \vec{u}_2, \ldots, \vec{u}_2 \} \quad \left( \text{or simply } \mathrm{Span}(\mathscr{L}) \right) \] \[ = \left\{ \begin{array}{l} \text{the collection of all the possible} \\ \text{lin. comb. of the } \vec{u}_1, \vec{u}_2, \ldots, \vec{u}_k \end{array} \right\} \] \[ = \{ \vec{v} : \vec{v} = c_1 \vec{u}_1 + c_2 \vec{u}_2 + \cdots + c_k \vec{u}_k \} \]

Linear Combination and Span

Given \( \mathscr{L} = \{ \vec{u}_1, \vec{u}_2, \ldots, \vec{u}_k \} \subseteq \mathbb{R}^n \).

Main Question :

Given a vector \( \vec{v} \in \mathbb{R}^n \), determine if

\[ \vec{v} \in \mathrm{Span}\{ \vec{u}_1, \ldots, \vec{u}_k \} \]

Find \( c_1, c_2, \ldots, c_k \) such that

\[ \vec{v} = c_1 \vec{u}_1 + c_2 \vec{u}_2 + \cdots + c_k \vec{u}_k \]

(General) Vector Space \( V \)

\( V \) is called a vector space as long as you can define vector addition and scalar mult. such that properties (1) — (8) hold.

  1. \[ \vec{u} + \vec{v} = \vec{v} + \vec{u} \]
  2. \[ (\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w}) \]
  3. \[ \alpha(\beta \vec{u}) = (\alpha\beta)\vec{u} \]
  4. \[ \alpha(\vec{u} + \vec{v}) = \alpha\vec{u} + \alpha\vec{v} \]
  5. \[ (\alpha + \beta)\vec{u} = \alpha\vec{u} + \beta\vec{u} \]

  6. There is \( \vec{0} \) such that

    \[ \vec{0} + \vec{u} = \vec{u} \]

  7. For each \( \vec{u} \), there is \( -\vec{u} \) such that

    \[ \vec{u} + (-\vec{u}) = \vec{0} \]
  8. \[ 1\vec{u} = \vec{u} \]

(General) Vector Space \( V \)

\( V \) is called a vector space as long as you can define vector addition and scalar mult. such that properties (1) — (8) hold.

(General) Vector Space \( V \)

Examples of (General) Vector Space

  1. \[ \begin{gathered} P_n = \{ \text{polynomials of degree at most } n \} \\[0.5em] = \{ p(x) \equiv a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \} \\[0.5em] p_1(x) \equiv a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \\[0.5em] p_2(x) \equiv b_0 + b_1 x + b_2 x^2 + \cdots + b_n x^n \\[0.5em] (p_1 + p_2)(x) = (a_0 + b_0) + (a_1 + b_1)x + \cdots + (a_n + b_n)x^n \end{gathered} \]

(General) Vector Space \( V \)

Examples of (General) Vector Space

  1. \[ \begin{gathered} P_n = \{ \text{polynomials of degree at most } n \} \\[0.5em] = \{ p(x) \equiv a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \} \\[0.5em] p_1(x) \equiv a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \\[0.5em] p_2(x) \equiv b_0 + b_1 x + b_2 x^2 + \cdots + b_n x^n \\[0.5em] (\alpha p_1)(x) \equiv \alpha a_0 + \alpha a_1 x + \alpha a_2 x^2 + \cdots + \alpha a_n x^n \end{gathered} \]

(General) Vector Space \( V \)

Examples of (General) Vector Space

  1. \[ \begin{gathered} \mathcal{F} = \{ \text{functions } f : \mathbb{R} \longrightarrow \mathbb{R} \} \\[0.5em] x \longrightarrow f(x) \\[0.5em] (f + g)(x) = f(x) + g(x) \\[0.5em] (\alpha f)(x) = \alpha f(x) \end{gathered} \]

(General) Vector Space \( V \)

Examples of (General) Vector Space

  1. \[ M^{m \times n} = \{ m \times n \text{ matrices} \} \]

    where \( m \) is the no. of rows and \( n \) is the no. of columns.

    \[ A = \begin{pmatrix} a_{11} & a_{12} & \cdots\cdots & a_{1n} \\ a_{21} & a_{22} & \cdots\cdots & a_{2n} \\ & \cdots\cdots & a_{ij} & \cdots\cdots \\ a_{m1} & a_{m2} & \cdots\cdots & a_{mn} \end{pmatrix} = (a_{ij})\]

(General) Vector Space \( V \)

Examples of (General) Vector Space

  1. \[ M^{m \times n} = \{ m \times n \text{ matrices} \} \]

    where \( m \) is the no. of rows and \( n \) is the no. of columns.

    \[ \begin{gathered} A = (a_{ij}), \quad B = (b_{ij}) \\[0.5em] (A + B) = (a_{ij}) + (b_{ij}) = (a_{ij} + b_{ij}) \\[0.5em] (\alpha A) = \alpha (a_{ij}) = (\alpha a_{ij}) \end{gathered} \]