please show all the properties, thank you. Let V = R? with the following vector addition and scalar multiplication:[X1 + y1 - 3]X2+ y2%3D(a) Is vector addition standard or non-standard?(b) Is scalar multiplication standard or non-standard?(c) If vector addition is non-standard, then what is the zero vector in V?(d) If vector addition is non-standard, then what do additive inverse (or opposite) looklike in V?(e) Determine if V is a vector space (Show all properties! If you don't have a vectorspace, then tell which properties hold and which ones don't!)

Name: please show all the properties, thank you. Let V = R? with the followi
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Description: please show all the properties, thank you. Let V = R? with the following vector addition and scalar multiplication:[X1 + y1 - 3]X2+ y2%3D(a) Is vector addition standard or non-standard?(b) Is scalar multiplication standard or non-standard?(c) If vect

Answered: Let V = R? with the following vector…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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please show all the properties, thank you.

Let V = R? with the following vector addition and scalar multiplication:
[X1 + y1 - 3]
X2+ y2
%3D
(a) Is vector addition standard or non-standard?
(b) Is scalar multiplication standard or non-standard?
(c) If vector addition is non-standard, then what is the zero vector in V?
(d) If vector addition is non-standard, then what do additive inverse (or opposite) look
like in V?
(e) Determine if V is a vector space (Show all properties! If you don't have a vector
space, then tell which properties hold and which ones don't!)

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Note: According to Bartleby guidelines; for more than three questions asked, only the first three are to be answered.

Given $V = ℝ^{2}$ with vector addition and scalar multiplication defined as:

$\begin{array}{rcl} [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] + [\begin{matrix} y_{1} \\ y_{2} \end{matrix}] & = & [\begin{matrix} x_{1} + y_{1} - 3 \\ x_{2} + y_{2} \end{matrix}] \\ c [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] & = & [\begin{matrix} c x_{1} \\ c x_{2} \end{matrix}] \end{array}$

Vector addition in $ℝ^{2}$ is said to be standard if $\begin{array}{rcl} [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] \end{array} \begin{array}{rcl} + \end{array} \begin{array}{rcl} [\begin{matrix} y_{1} \\ y_{2} \end{matrix}] \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} [\begin{matrix} x_{1} + y_{1} \\ x_{2} + y_{2} \end{matrix}] \end{array}$

Vector multiplication in $ℝ^{2}$ is said to be standard if $\begin{array}{rcl} c \end{array} \begin{array}{rcl} [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} [\begin{matrix} c x_{1} \\ c x_{2} \end{matrix}] \end{array}$

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