MATH347

MATH347 L1: Linear combinations

Lesson concepts:

Vectors

Some quantities arising in applications can be expressed as single numbers, called “scalars”
- Speed of a car on a highway $v = 35$ mph
- A person's height $H = 183$ cm
Many other quantitites require more than one number:
- Position in a city: “Intersection of 86^{$t h$} St and 3^{$r d$} Av”
- Position in 3D space: $(x, y, z)$
- Velocity in 3D space: $(u, v, w)$

Vector definition

Definition. A vector is a grouping of $m$ scalars

$𝒗 = [\begin{array}{c} v_{1} \\ v_{2} \\ ⋮ \\ v_{m} \end{array}] \in S^{m}, v_{i} \in S$

The scalars usually are naturals ( $S = N$ ), integers ( $S = Z$ ), rationals ( $S = Q$ ), reals ( $S = R$ ), or complex numbers ( $S = C$ )
We often denote the dimension and set of scalars as $𝒗 \in S^{m}$ , e.g. $𝒗 \in R^{m}$

Sets of vectors are denoted as

𝒱 = {𝒗 = [\begin{array}{c} v_{1} \\ v_{2} \\ ⋮ \\ v_{m} \end{array}], v_{i} \in S}

(1)

A vector can also be interpreted as a function from a subset of $ℕ$ to $S$
$v : {1, 2, . ., m} \to S$

Vector operations

Vector addition. Consider two vectors $𝒖, 𝒗 \in 𝒱$ . We define the sum of the two vectors as the vector containing the sum of the components
$𝒘 = 𝒖 + 𝒗 = [\begin{array}{c} u_{1} \\ u_{2} \\ ⋮ \\ u_{m} \end{array}] + [\begin{array}{c} v_{1} \\ v_{2} \\ ⋮ \\ v_{m} \end{array}] = [\begin{array}{c} u_{1} + v_{1} \\ u_{2} + v_{2} \\ ⋮ \\ u_{m} + v_{m} \end{array}] = [\begin{array}{c} w_{1} \\ w_{2} \\ ⋮ \\ w_{m} \end{array}]$
>>

u=[1 2 3]; v=[-2 1 2]; u+v
$(\begin{array}{ccc} - 1 & 3 & 5 \end{array})$
Scalar multiplication. Consider $α \in S$ , $𝒖 \in 𝒱$ . We define the multiplication of vector $𝒖$ by scalar $α$ as the vector containing the product of each component of $𝒖$ with the scalar $α$
$𝒘 = α 𝒖 = [\begin{array}{c} α u_{1} \\ α u_{2} \\ ⋮ \\ α u_{m} \end{array}] = [\begin{array}{c} w_{1} \\ w_{2} \\ ⋮ \\ w_{m} \end{array}]$

Linear combinations

Linear combination. Let $α, β \in S$ , $𝒖, 𝒗 \in 𝒱$ . Define a linear combination of two vectors by
$𝒘 = α 𝒖 + β 𝒗 = [\begin{array}{c} α u_{1} \\ α u_{2} \\ ⋮ \\ α u_{m} \end{array}] + [\begin{array}{c} β v_{1} \\ β v_{2} \\ ⋮ \\ b β v_{m} \end{array}] = [\begin{array}{c} α u_{1} + β v_{1} \\ α u_{2} + β v_{2} \\ ⋮ \\ α u_{m} + β v_{m} \end{array}] = [\begin{array}{c} w_{1} \\ w_{2} \\ ⋮ \\ w_{m} \end{array}]$
Linear combination of $n$ vectors
$𝒃 = x_{1} 𝒂_{1} + x_{2} 𝒂_{2} + \dots + x_{n} 𝒂_{n} = [\begin{array}{l} x_{1} a_{11} + x_{2} a_{12} + \dots + x_{n} a_{1 n} \\ x_{1} a_{21} + x_{2} a_{22} + \dots + x_{n} a_{2 n} \\ ⋮ \\ x_{1} a_{m 1} + x_{2} a_{m 2} + \dots + x_{n} a_{m n} \end{array}]$

Uses of linear combinations

"Start at the center of town. Go east 3 blocks and north 2 blocks. What is your final position?"

𝒔 = [\begin{array}{c} 0 \\ 0 \end{array}], 𝒆_{E} = [\begin{array}{c} 1 \\ 0 \end{array}], 𝒆_{N} = [\begin{array}{c} 0 \\ 1 \end{array}]

α_{E} = 3, α_{N} = 2

𝒇 = 𝒔 + α_{E} 𝒆_{E} + α_{N} 𝒆_{N} = [\begin{array}{c} 0 \\ 0 \end{array}] + 3 [\begin{array}{c} 1 \\ 0 \end{array}] + 2 [\begin{array}{c} 0 \\ 1 \end{array}] = [\begin{array}{c} 3 \\ 2 \end{array}]

Linear combinations allow us to express a position in space using a standard set of directions. Questions:

Matrices

Seek a more compact notation for the linear combination

a [\begin{array}{c} 1 \\ 2 \\ 3 \end{array}] + b [\begin{array}{c} - 1 \\ 0 \\ 1 \end{array}] + c [\begin{array}{c} 2 \\ 0 \\ 1 \end{array}] = [\begin{array}{c} a - b + 2 c \\ 2 a \\ 3 a + b + c \end{array}]

Group the vectors together to form a “matrix”
$𝑨 = [\begin{array}{ccc} 1 & - 1 & 2 \\ 2 & 0 & 0 \\ 3 & 1 & 1 \end{array}]$
Group the scalars together to form a vector
$𝒖 = [\begin{array}{c} a \\ b \\ c \end{array}]$

Matrix-vector multiplication = linear combination

Define matrix-vector multiplication
$𝑨 𝒖 = [\begin{array}{ccc} 1 & - 1 & 2 \\ 2 & 0 & 0 \\ 3 & 1 & 1 \end{array}] [\begin{array}{c} a \\ b \\ c \end{array}] = [\begin{array}{c} a - b + 2 c \\ 2 a \\ 3 a + b + c \end{array}]$
In general
$𝑨 = [\begin{array}{llll} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{array}] = [\begin{array}{llll} 𝒂_{1} & 𝒂_{2} & \dots & 𝒂_{n} \end{array}], 𝒙 = [\begin{array}{l} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{array}]$ $𝒃 = 𝑨 𝒙 = x_{1} 𝒂_{1} + x_{2} 𝒂_{2} + \dots + x_{n} 𝒂_{n} = [\begin{array}{l} x_{1} a_{11} + x_{2} a_{12} + \dots + x_{n} a_{1 n} \\ x_{1} a_{21} + x_{2} a_{22} + \dots + x_{n} a_{2 n} \\ ⋮ \\ x_{1} a_{m 1} + x_{2} a_{m 2} + \dots + x_{n} a_{m n} \end{array}]$

Examples: linear combination

Examples: linear combination by matrix-vector combination