Give a correct answer for given question (A) Argue why { (1,0,3), (2,3,1), (0,0,1) } is a coordinate system ( bases ) for R^3 ?(B) Find the coordinates of (7, 6, 16) relative to the set in part (A)

Dottie Parra 2021-01-23 Answered

Give a correct answer for given question

(A) Argue why \({ (1,0,3), (2,3,1), (0,0,1) }\) is a coordinate system ( bases ) for \(R^3\) ?

(B) Find the coordinates of \((7, 6, 16)\) relative to the set in part (A)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

Adnaan Franks
Answered 2021-01-24 Author has 6853 answers

Given \(B = \left\{ \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 3\\ 1 \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix} \right\}\)

By definition, we say that tha vector \(\left\{ v_1, v_2, ..., v_n \right\}\) are linearly dependent if there exists scalars \(\displaystyle\alpha_{{1}},\alpha_{{2}},\ldots,\alpha_{{n}}\) not all of them zero such that \(\displaystyle\alpha_{{1}}{v}_{{1}}+\alpha_{{2}}{v}_{{2}}+\ldots+\alpha_{{n}}{v}_{{n}}=\overline{{0}}.\)

We say that the vectors \(\left\{ v_1, v_2, ..., v_n \right\}\) are linearly independent if \(\displaystyle\alpha_{{1}}{v}_{{1}}+\alpha_{{2}}{v}_{{2}}+\ldots+\alpha_{{n}}{v}_{{n}}=\overline{{0}}\) then  

The set of vectors \(\left\{ v_1, v_2, ..., v_n \right\}\) is said to be a basis for vector space V if i) set of vectors \(\left\{ v_1, v_2, ..., v_n \right\}\) is linearly independent ii) span \(\left\{ v_1, v_2, ..., v_n \right\}=V\) If B = \(\left\{ v_1, v_2, ..., v_n \right\}\) is a basis in a vector space V than every vector \(\overrightarrow{v}\in{V}\) can be uniquely expressed as a linear combination of basis vectors \(\displaystyle{b}_{{1}},{b}_{{2}},\ldots,{b}_{{n}}.\) i.e there exists unique scalsrs \(\displaystyle\alpha_{{1}},\alpha_{{2}},\ldots,\alpha_{{n}}\) such that, \(\overrightarrow{v}=\alpha_{{1}}{b}_{{1}}+,\alpha_{{2}}{b}_{{2}}+\ldots+\alpha_{{n}}{b}_{{n}}.\) The coordinates of the vector \(\overrightarrow{v}\) relative to the basis \(\displaystyle{B}\) is the sequence of co-ordinates, i.e. \([v]_B = (\alpha_1, \alpha_2, .., \alpha_n)\) 

Consider \(A = \begin{bmatrix} 1 & 2 & 0 \\ 0 & 3 & 0 \\ 3 & 1 & 1 \end{bmatrix}\) Applying \(\displaystyle{R}_{{3}}\rightarrow{R}_{{3}}-{3}{R}_{{1}},\) we get
\(R_3 \rightarrow R_3 - 3 \ R_1,\ we \ get\ A \sim \begin{bmatrix} 1 & 2 & 0 \\ 0 & 3 & 0 \\ 0 & -5 & 1 \end{bmatrix}\)

Applying \(\displaystyle{R}_{{3}}\rightarrow{5}{R}_{{2}}+{3}{R}_{{3}},\) we get \(\sim \begin{bmatrix} 1 & 2 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}\) (because in eacelon form, first, second and third columns have pivot elements)

\(\displaystyle\therefore\) The set of vectors \(\left\{ \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 3\\ 1 \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix} \right\}\) is linearly independent dim \((R^3)=3 \ dim\ (\left\{ \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 3\\ 1 \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix} \right\}) = 3 \ \Rightarrow \left\{ \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 3\\ 1 \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix} \right\} = R^3 \ \)

\(\therefore\)Basis for \(R^3\ is \left\{ \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 3\\ 1 \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix} \right\}\) b. Let \(\begin{bmatrix}7 \\ 6 \\ 16 \end{bmatrix} = (a)\begin{bmatrix}1 \\ 0 \\ 3 \end{bmatrix} + (b) \begin{bmatrix}2 \\ 3 \\ 1 \end{bmatrix} + (c) \begin{bmatrix}0 \\ 0 \\ 1 \end{bmatrix} \ \Rightarrow a + 2b = 7 \rightarrow (i) \ 3b = 6 \ \Rightarrow b = 2 \rightarrow (ii) \ 3a + b + c = 16 \rightarrow (iii)\)

From (i), \(\displaystyle{a}={7}-{2}{b}={7}-{4}={3}.\)

From (ii), \(3a + b + c = 16 \ \Rightarrow c = 16 - 3a - b = 16 - 9 - 2 = 5 \ \therefore \overrightarrow{v} = \begin{bmatrix}7 \\ 6 \\ 16 \end{bmatrix} = (3)\begin{bmatrix}1 \\ 0 \\ 3 \end{bmatrix} + (2)\begin{bmatrix}2 \\ 3 \\ 1 \end{bmatrix} + (5)\begin{bmatrix}0 \\ 0 \\ 1 \end{bmatrix} \ \)

\(\therefore\)The coordinates of the vector \(\overrightarrow{v} = \begin{bmatrix}7 \\ 6 \\ 16 \end{bmatrix}\) raletive to the basis \(B = \left\{ \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0\\ 1 \end{bmatrix} \right\} is [\overrightarrow{v}]_B = (a, b, c) = (3, 3, 5)\)

Have a similar question?
Ask An Expert
13
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-08-01
How are triple integrals defined in cylindrical and spherical coor-dinates? Why might one prefer working in one of these coordinate systems to working in rectangular coordinates?
asked 2021-01-02

Solve the given Alternate Coordinate Systems and give a correct answer 10) Convert the equation from Cartesian to polar coordinates solving for \(r^2\):
\(\frac{x^2}{9} - \frac{y^2}{16} = 25\)

asked 2021-06-05
The Cartesian coordinates of a point are given.
a) \((2,-2)\)
b) \((-1,\sqrt{3})\)
Find the polar coordinates \((r,\theta)\) of the point, where r is greater than 0 and 0 is less than or equal to \(\theta\), which is less than \(2\pi\)
Find the polar coordinates \((r,\theta)\) of the point, where r is less than 0 and 0 is less than or equal to \(\theta\), which is less than \(2\pi\)
asked 2021-08-08
Discuss: Different Coordinate Systems As was noted in the overview of the chapter, certain curves are more naturally described in one coordinate system than in another. In each of the following situations, which coordinate system would be appropriate: rectangular or polar? Give reasons to support your answer.
a) You need to give directions to your house to a taxi driver.
b) You need to give directions to your house to a homing pigeon.
asked 2021-08-02
Given point \(\displaystyle{P}{\left(-{2},{6},{3}\right)}\) and vector \(\displaystyle{B}={y}{a}_{{{x}}}+{\left({x}+{z}\right)}{a}_{{{y}}}\), Express P and B in cylindrical and spherical coordinates. Evaluate A at P in the Cartesian, cylindrical and spherical systems.
asked 2021-08-03
a) What coordinate system is suggested to be used when seeing the integrand \(\displaystyle{\left({x}^{{{2}}}+{y}^{{{2}}}\right)}^{{\frac{{5}}{{2}}}}\), cylindrical or Cartesian?
b) List the conversion factors for rectangular to cylindrical coordinate systems.
asked 2021-08-04
Solving Systems Graphically- One Solution
Find or create an example of a system of equations with one solution.
Graph and label the lines on a coordinate plane. Provide their equations.
State the accurate solution to the system.
...