9.3: Vector spaces in R^n

Last updated
Save as PDF

Page ID: 64303

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

There are two properties that define a vector space these are:

Closed under addition
Closed under scalar multiplication

For now we will consider vector spaces in \(R^n\) which are just vectors of real numbers (ex: [10,20,3.2], [5,8,32], [8,-0.7], etc) where \(n\) is just the length of the vector (ex: 3, 3, and 2 in the earlier example). In the general case a vector does not have to be composed of real numbers but can be almost any type of object as long as it maintains the two above properties, we will get into this concept later in the semester. In the case of real number the above concepts can be described as follows:

Closed under addition means that if we add any two real vectors vectors (i.e. \(u\), \(v \in R^n\)) then the result is also in \(R^n\)). This is easy to understand if you think about adding any two real vectors there is no way to get a result that is not also a real vector. A way to say this mathematically is as follows:

\[\text{if } u,v \in R^n \nonumber \]

\[\text{then } u+v \in R^n \nonumber \]

Closed under scalar multiplication means that if we have any scalar number (\(s \in R\)) and we multiply it by a real vector (\(v \in R^n\)) then the result is also a vector in \(R^n\). Since multiplying a real number by a real number results in a real number this one is also true. Or we can say it as follows:

\[\text{if } s \in R \text{ and } v \in R^n \nonumber \]

\[\text{then } sv \in R^n \nonumber \]

The following are some properties of vector addition and multiplication for vectors \(u\) and \(v\):

\(u + v = v + u\) Commutative property
\(u + (v + w) = (u + v) + w\) Associative property
\(u+0 = 0 + u = u\) Property of zero vector
\(u + (-u) = 0\) Property of the negative vector
\(c(u+v) = cu + cv\) Distributive properties
\((c+d)u = cu+du\) Distributive Properties
\(c(du) = (cd)u\) Distributed Properties
\(1u = u\) Scalar multiplication by 1

Question

Compute the following linear combinations for \(u=(1,2)\), \(v=(4,−1)\), and \(w=(−3,5)\).

(a) \(a = u+w\)

# Put your answer here

from answercheck import checkanswer

checkanswer.vector(a,'af464d466ae982f2cd4461af494e86d6');

(b) \(a = 2u+v\)

# Put your answer here

from answercheck import checkanswer

checkanswer.vector(a,'393468eff8c6ba5d27b7d0aa1b18f929');

(c) \(a = u+3w\)

# Put your answer here

from answercheck import checkanswer

checkanswer.vector(a,'d5e5ca43a86501bcde09b1cbc0ba49b5');