For the given set of all vectors shown as $ W=\left[ \begin{matrix}4a\ +\ 3b\\0\\ \begin{matrix}a+b+c\\c\ -\ 2a\\\end{matrix}\\\end{matrix}\right] $, and here a, b and c are arbitrary real numbers. Find vector set S which spans W or give an example to show that W is not a space vector.
In this question, we have to find a set S, which spans the given set of all vectors W.
Vector
The basic concept to solve this question requires that we should have sound knowledge of vector space and arbitrary real values.
The arbitrary values in a matrix can be any value belonging to real numbers.
In mathematics, a Vector space is defined as a non-empty set that full fills the following 2 conditions:
- Addition $ u+v = v+u $
- Multiplication by real numbers
Sum of vector
Multiplication of vector
Expert Answer
In the question, we are given the set of all vectors $W$ which is written as follows:
\[ \left[ \begin{matrix} 4a\ +\ 3b\\0\\ \begin{matrix}a+b+c\\c\ -\ 2a\\ \end{matrix}\\ \end{matrix} \right ] \]
From the given set, we can write that:
\[ a =\left[ \begin{matrix} 4\\0\\ \begin{matrix} 1\\-\ 2\\ \end{matrix}\\ \end{matrix} \right] \]
\[ b\ =\left[ \begin{matrix} \ 3\\0\\ \begin{matrix} 1\\0\\ \end{matrix}\\ \end{matrix} \right] \]
\[ c\ = \left[\begin{matrix} \ 0\\0\\ \begin{matrix} 1\\ 1\\ \end{matrix}\\ \end{matrix} \right] \]
So the required equation becomes as follows:
\[ w= a \left[ \begin{matrix} 4\\0\\ \begin{matrix}1\\-\ 2\\ \end{matrix}\\ \end{matrix} \right]\ +b\ \left[ \begin{matrix} \ 3\\0\\ \begin{matrix}1\\0\\ \end{matrix} \\ \end{matrix} \right]\ +c\ \left[ \begin{matrix}\ 0\\0\\ \begin{matrix}1\\1\\ \end{matrix}\\ \end{matrix} \right] \]
We can write it as the set of all vectors in terms of the set $S$:
\[ S = \left[\begin{matrix} 4\\0\\ \begin{matrix}1\\-\ 2\\\end{matrix}\\\end{matrix} \right]\ ,\ \left[ \begin{matrix} \ 3\\0\\\begin{matrix} 1\\0\\ \end{matrix}\\\end{matrix} \right]\ ,\ \left[\begin{matrix}\ 0\\0\\\begin{matrix} 1\\1\\ \end{matrix}\\ \end{matrix}\right] \]
So our required equation is as follows:
\[ S=\ \left\{\ \left[ \begin{matrix} 4\\0\\\begin{matrix} 1\\-\ 2\\\end{matrix}\\\end{matrix}\right]\ ,\ \left[ \begin{matrix} \ 3\\0\\ \begin{matrix} 1\\0\\ \end{matrix}\\ \end{matrix} \right]\ ,\ \left[ \begin{matrix}\ 0\\0\\\begin{matrix} 1\\1\\ \end{matrix} \\\end{matrix} \right]\ \ \right\} \]
Numerical Results
Our required set of $S$ with all vector equations is as follows:
\[ S=\ \left\{\ \left[ \begin{matrix} 4\\0\\\begin{matrix} 1\\-\ 2\\\end{matrix}\\\end{matrix}\right]\ ,\ \left[ \begin{matrix} \ 3\\0\\ \begin{matrix} 1\\0\\ \end{matrix}\\ \end{matrix} \right]\ ,\ \left[ \begin{matrix}\ 0\\0\\\begin{matrix} 1\\1\\ \end{matrix} \\\end{matrix} \right]\ \ \right\} \]
Example
For the given set of all vectors shown as $ W= \left[ \begin{matrix} -2a\ +\ 3b\ \\-7c\\ \begin{matrix} a+b+c\\c\ \\ \end{matrix}\\ \end{matrix} \right] $, and here $a$, $b$ and $c$ are arbitrary real numbers. Find vector set $S$ which spans $W$ or give an example to show that $W$ is not a space vector.
Solution
Given the matrix, we have:
\[ \left[\begin{matrix}-2a\ +\ 3b\ \\-7c\\\begin{matrix}a+b+c\\c\ \\\end{matrix}\\\end{matrix}\right] \]
From the given set, we can write that:
\[ a=\left[\begin{matrix}-2\\0\\\begin{matrix}1\\0\\\end{matrix}\\\end{matrix}\right] \]
\[ b\ =\left[\begin{matrix}\ 3\\0\\\begin{matrix}1\\0\\\end{matrix}\\\end{matrix}\right] \]
\[ c\ =\left[\begin{matrix}\ 0\\-7\\\begin{matrix}1\\1\\\end{matrix}\\\end{matrix}\right] \]
So, the required equation becomes:
\[ W=a\left[\begin{matrix}-2\\0\\\begin{matrix}1\\0\\\end{matrix}\\\end{matrix}\right]\ +b\ \left[\begin{matrix}\ 3\\0\\\begin{matrix}1\\0\\\end{matrix}\\\end{matrix}\right]\ +c\ \left[\begin{matrix}\ 0\\-7\\\begin{matrix}1\\1\\\end{matrix}\\\end{matrix}\right] \]
We can also write it as follows:
\[ S=\left[\begin{matrix}-2\\0\\\begin{matrix}1\\0\\\end{matrix}\\\end{matrix}\right]\ ,\ \left[\begin{matrix}\ 3\\0\\\begin{matrix}1\\0\\\end{matrix}\\\end{matrix}\right]\ ,\ \left[\begin{matrix}\ 0\\-7\\\begin{matrix}1\\1\\\end{matrix}\\\end{matrix}\right] \]
Our required set of $S$ with all the vector equations is as follows:
\[ S=\ \left\{\ \left[\begin{matrix}-2\\0\\\begin{matrix}1\\0\\\end{matrix}\\\end{matrix}\right]\ ,\ \left[\begin{matrix}\ 3\\0\\\begin{matrix}1\\0\\\end{matrix}\\\end{matrix}\right]\ ,\ \left[\begin{matrix}\ 0\\-7\\\begin{matrix}1\\1\\\end{matrix}\\\end{matrix}\right]\ \ \right\} \]