Introduction to Vector Spaces - Revision history

Khanh at 16:45, 4 November 2021

2021-11-04T16:45:44Z

Lila at 18:55, 29 September 2021

2021-09-29T18:55:22Z

Lila at 18:50, 29 September 2021

2021-09-29T18:50:04Z

Lila at 18:45, 29 September 2021

2021-09-29T18:45:10Z

Lila: Created page with "A '''vector space''' (over $\R$ ) consists of a set $V$ along with two operations " $+$ " and " $\cdot$ " subject to these conditions...."

2021-09-29T18:36:55Z

Created page with "A '''vector space''' (over <math>\R</math>) consists of a set <math>V</math> along with two operations "<math>+</math>" and "<math>\cdot</math>" subject to these conditions...."

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A '''vector space''' (over <math>\R</math>) consists of a set <math>V</math> along with
two operations "<math>+</math>" and "<math>\cdot</math>" subject to these conditions.

# For any <math>\vec v,\vec w\in V:\vec v+\vec w\in V</math>.
# For any <math>\vec v,\vec w\in V:\vec v+\vec w=\vec w+\vec v</math>.
# For any <math>\vec u,\vec v,\vec w\in V:(\vec v+\vec w)+\vec u=\vec v+(\vec w+\vec u)</math>.
# There is a '''zero vector''' <math>\vec0\in V</math> such that <math>\vec v+\vec0=\vec v</math> for all <math>\vec v\in V</math>.
# Each <math>\vec v\in V</math> has an '''additive inverse''' <math>\vec w\in V</math> such that <math>\vec w+\vec v=\vec0</math>.
# If <math>r</math> is a '''scalar''', that is, a member of <math>\R</math> and <math>\vec v\in V</math> then the '''scalar multiple''' <math>r\cdot\vec v </math> is in <math>V</math>.
# If <math>r,s\in\R</math> and <math>\vec v\in V</math> then <math>(r+s)\cdot\vec v=r\cdot\vec v+s\cdot\vec v</math>.
# If <math>r\in\R</math> and <math>\vec v,\vec w\in V</math> , then <math>r\cdot(\vec v+\vec w)=r\cdot\vec v+r\cdot\vec w</math>.
# If <math>r,s\in\R</math> and <math>\vec v\in V</math> , then <math>(rs)\cdot\vec v=r\cdot(s\cdot\vec v)</math>.
# For any <math>\vec v\in V</math> , <math>1\cdot\vec v=\vec v</math>.

Remark: Because it involves two kinds of addition and two kinds of multiplication, that definition may seem confused. For instance, in condition 7 "<math>(r+s)\cdot\vec v=r\cdot\vec v+s\cdot\vec v</math>", the first "<math>+</math>" is the real number addition operator while the "<math>+</math>" to the right of the equals sign represents vector addition in the structure <math>V</math> . These expressions aren't ambiguous because, e.g., <math>r</math> and <math>s</math> are real numbers so "<math>r+s</math>" can only mean real number addition.

← Older revision		Revision as of 16:45, 4 November 2021
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	==Resources==		==Resources==
−	* [https://en.wikibooks.org/wiki/Linear_Algebra/Definition_and_Examples_of_Vector_Spaces Definition and Examples of Vector Spaces], WikiBooks
	* [https://www.math.uh.edu/~jiwenhe/math2331/lectures/sec4_1.pdf Linear Algebra: Vector Spaces and Subspaces], University of Houston		* [https://www.math.uh.edu/~jiwenhe/math2331/lectures/sec4_1.pdf Linear Algebra: Vector Spaces and Subspaces], University of Houston
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Linear_Algebra/Definition_and_Examples_of_Vector_Spaces Definition and Examples of Vector Spaces, WikiBooks] under a CC BY-SA license

← Older revision		Revision as of 18:55, 29 September 2021
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	inherited from the space in the prior example. (We can think of <math> F </math> as "the same" as <math> \mathbb{R}^2 </math> in that <math>a\cos\theta+b\sin\theta</math> corresponds to the vector with components <math>a</math> and <math>b</math>.)		inherited from the space in the prior example. (We can think of <math> F </math> as "the same" as <math> \mathbb{R}^2 </math> in that <math>a\cos\theta+b\sin\theta</math> corresponds to the vector with components <math>a</math> and <math>b</math>.)
		+
		+
		+	==Resources==
		+	* [https://en.wikibooks.org/wiki/Linear_Algebra/Definition_and_Examples_of_Vector_Spaces Definition and Examples of Vector Spaces], WikiBooks
		+	* [https://www.math.uh.edu/~jiwenhe/math2331/lectures/sec4_1.pdf Linear Algebra: Vector Spaces and Subspaces], University of Houston

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 Remark: Because it involves two kinds of addition and two kinds of multiplication, that definition may seem confused. For instance, in condition 7 "<math>(r+s)\cdot\vec v=r\cdot\vec v+s\cdot\vec v</math>", the first "<math>+</math>" is the real number addition operator while the "<math>+</math>" to the right of the equals sign represents vector addition in the structure <math>V</math> . These expressions aren't ambiguous because, e.g., <math>r</math> and <math>s</math> are real numbers so "<math>r+s</math>" can only mean real number addition.
-Example:
+Lemma 1.17: In any vector space <math> V </math>, for any <math> \vec{v}\in V </math> and <math> r\in\mathbb{R} </math>, we have
 The set <math>\R^2</math> is a vector space if the operations "<math> + </math>" and "<math> \cdot </math>" have their usual meaning.
@@ Line 72: / Line 92: @@
 In a similar way, each <math> \mathbb{R}^n </math> is a vector space with the usual operations of vector addition and scalar multiplication. (In <math> \mathbb{R}^1 </math>, we usually do not write the members as column vectors, i.e., we usually do not write "<math> (\pi) </math>". Instead we just write "<math> \pi </math>".)