ENGG2013Unit 4 Checking out at sushi bar

Jan, 2011.

Row and Column Vectors

• column vector

• row vector

• n-dimensionalcolumn vector

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n

A list of n numbers written vertically

Convention: a vector isby default a column vectorin ENGG2013.

Convention: The componentsin a vector are sometimecalled “scalar”.

Illustration

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x

y

(0,0)

(5,3)

5

3

(4,4)

(1,-1)

Notation using column vector

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x

y

5

3

Notation from Physics

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x

y

zhas the same meaning as

Interpretation of vector (I)

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y

(5,3)

5

3

location

or

Interpretation of vector (2)

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y

(0,0)

(5,3)

5

3

an arrow from the origin

or

Interpretation of vector (3)

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y

(0,0)

(5,3)

5

3

Any arrow in the same directionwith the same length

or

Mathematical Notation

• The set of all 2-D vectors with real numbers as components is denoted by

• The set of all 3-D vectors with real numbers as components is denoted by

• The set of all n-D vectors with real numbers as components:

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Equality for vectors...

• … is just equality in each component• Examples

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Vector addition …

• … is just component-wise addition

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But has no meaning.

Scalar Multiplication

• Multiply each component by the scalar constant.• Interpreted as lengthening, or shortening the

vector, but keeping the same direction.

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x

y

z

Dot product

• A.k.a. scalar product, or inner product.• For 2-D vector,

• It measures the “angle” between two vectors.– The dot product of two vectors is zero if the two

vectors are perpendicular

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Dot product in general

• For n-dimensional vectors in general, we define the dot product as

• Example

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Two n-dim vectors aresaid to be perpendicular,or orthogonal, if their dot productis equal to 0.

Simple properties

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For any two vectors u, v and w of the same dimension, and constants c and d.

Matrix-vector multiplication

• Given an mn matrix A, and an n-dimensional vector x, the product of A and x

is an m-dimensional vector defined as

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For double subscripts,the first subscript is therow index and the secondis the column index

Just compute dot products m times

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Dot product of the first row in the matrix and the column vector

Dot product of the second row in the matrix and the column vector

Dot product of the last row in the matrix and the column vector

Example

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is un-defined.

2x4 3x1

2x4 4x12x1

Checking out in a sushi bar• Four prices

– Red plate: $10– Blue plate: $20– Green plate: $35– Pink plate: $50

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Customer A

Customer B

R B G P

Nutrition problem

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In matrix notation:

Amounts offood A, B,C and D

Requirementsof the fournutrients

(protein)(carbohydrate)

(vitamin A)

(vitamin C)

Vector equation

• Equation involving vectors.• E.g. Find a and b such that

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The nutrition problem asvector equation

• Just another way to write the same thing.kshum ENGG2013 22

Four different perspectives

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A system of linear equations Ax = b

Augmented matrixVector equation