Ch 1A - Rn and Cn

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Introduction to vector spaces Rn and Cn. Do not test on multiple choice or true or false.

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29 Terms

1
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The study of linear maps on finite-dimensional vector spaces Rn:

Linear algebra

2
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A set with arithmetic rules that satisfy natural algebraic properties:

Vector space

3
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The set of all complex numbers C is denoted by the expression:

a+bi; a,b in R

4
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What set is denoted by the expression “α + βἱ: α,β in R?

Complex numbers C

5
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“α + β = β + α” is what property of complex arithmetic?

commutativity

6
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“(α + β) + λ = α + (β + λ )” and “(α β) λ = α (β λ)” are what property of complex arithmetic?

associativity

7
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“λ + 0 = λ” is what property of complex arithmetic?

additive identity

8
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“λ 1 = λ” is what property of complex arithmetic?

multiplicative identity

9
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For every α in C, there exists a unique β in C such that “α + β = 0”. What property of complex arithmetic is this?

additive inverse

10
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For every α in C with α ≠ 0, there exists a unique β in C such that “α β = 1”. What is this property of complex arithmetic?

multiplicative inverse

11
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“λ (α + β) = λ α + λ β”, where all variables are complex numbers. What is this property of complex arithmetic?

distributive property

12
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What does the vector space F stand for?

Either R or C

13
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When emphasizing that an object is a number, not a vector, we call it an element of F. What is another term for this number?

scalar

14
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<p>A ____ of length <em>n</em> is an ordered collection of <em>n</em> elements</p>

A ____ of length n is an ordered collection of n elements

list

15
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It is the number of elements within a list

length

16
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What do we call elements separated by commas and surrounded by parentheses?

list

17
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T or F: A list may have infinite length.

F

18
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T or F: Order matters and repetitions have meaning for lists

T

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T or F: sets work the same as lists, the only difference is their annotation

F

20
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T or F: For sets, order and repetitions are relevant

F

21
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Consider the set x in Fn, what does “xk” stand for? (where κ is a number from 1 to n)

kth coordinate from x1 to xn

22
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It is the kth position within a list x in Fn, a number from 1 to n.

coordinate

23
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<p><strong>T or F:</strong> Two lists are equal even if they do not share the same length or the elements are out of order</p>

T or F: Two lists are equal even if they do not share the same length or the elements are out of order

F

<p>F</p>
24
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<p>It is defined by adding the corresponding <strong>k</strong>th coordinate</p>

It is defined by adding the corresponding kth coordinate

Addition in Fn

<p>Addition in Fn</p>
25
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If x, y are elements of Fn, then “x + y = y + x”. What is the name of this property?

commutativity of addition in Fn

26
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<p>What does this illustration help visualize?</p>

What does this illustration help visualize?

The sum of two vectors

<p>The sum of two vectors</p>
27
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<p>For <strong>x</strong> in <strong>Fn</strong>, there exists a vector <strong>-x</strong> (in <strong>Fn</strong>) such that <strong>x + (-x) = 0</strong>. What is this the definition to?</p>

For x in Fn, there exists a vector -x (in Fn) such that x + (-x) = 0. What is this the definition to?

additive inverse in Fn

<p>additive inverse in Fn</p>
28
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<p>__________ is defined as multiplying each coordinate of a vector <strong>x</strong> by a scalar λ</p>

__________ is defined as multiplying each coordinate of a vector x by a scalar λ

scalar multiplication in Fn

<p>scalar multiplication in Fn</p>
29
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It is a set containing at least two distinct elements called 0 or 1, along with operations of addition and multiplication satisfying all properties of complex arithmetic. Examples of it are C and R.

field