Answer:
[tex]b[/tex] isn't a linear combination of the vectors formed from the columns of the matrix [tex]A.[/tex]
Step-by-step explanation:
The question is misspelled. The correct [tex]A[/tex] matrix and [tex]b[/tex] vector are :
[tex]A=\left[\begin{array}{ccc}1&-4&2\\0&3&5\\-2&8&-4\end{array}\right][/tex]
[tex]b=\left[\begin{array}{c}3&-7&-3\end{array}\right][/tex]
In order to determine if [tex]b[/tex] is a linear combination of the vectors formed from the columns of the [tex]A[/tex] matrix we write the following linear combination with the columns from [tex]A[/tex] :
[tex]\left[\begin{array}{c}1&0&-2\end{array}\right]x_{1}+\left[\begin{array}{c}-4&3&8\end{array}\right]x_{2}+\left[\begin{array}{c}2&5&-4\end{array}\right]x_{3}=\left[\begin{array}{c}3&-7&-3\end{array}\right][/tex] (I)
This can be also written as :
[tex]AX=b[/tex]
Where [tex]X[/tex] is the following vector : [tex]X=\left[\begin{array}{c}x_{1}&x_{2}&x_{3}\end{array}\right][/tex]
The matrix from the system (I) is :
[tex]\left[\begin{array}{cccc}1&-4&2&3\\0&3&5&-7\\-2&8&-4&-3\\\end{array}\right][/tex]
Applying matrix operations to the system matrix we obtain this equivalent matrix :
[tex]\left[\begin{array}{cccc}1&-4&2&3\\0&3&5&-7\\0&0&0&3\end{array}\right][/tex]
From the third row we obtain that
[tex]0x_{1}+0x_{2}+0x_{3}=3[/tex]
Which is an absurd. Therefore the system has no solution and it doesn't exist a linear combination of the vectors formed from the columns of the matrix [tex]A[/tex] to obtain the vector [tex]b[/tex].
Finally, [tex]b[/tex] isn't a linear combination of the vectors formed from the columns of the matrix [tex]A.[/tex]
