Let $e_1 = (1, 1, 2, 1)$; $e_2 = (0, 2, 3, 1)$, $e_3 = (-2, 3, 1, 1)$ and $e_4 = (-6, 4, 2, -1)$. I need to show that the system $\{e_1, e_2, e_3, e_4\}$ is a base of vectorspace $\mathbb{R}^4$. I also need to find coordinates of $x = (3,2,7,2)$ on that base. I don't even know how to begin.
Find a base of vectorspace
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vector-spaces
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0Do you know about determinants? Gauß Jordan? – 2017-02-27
2 Answers
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how to begin:
you have to prove that $e_1, e_2, e_3, e_4$ are linearly independent. It can be checked for example by checking if the matrix made by these vectors is invertible.
coordinates of $x$ in this base are (if exists) numbers $a,b,c,d$ such that $x=ae_1+be_2+ce_3+de_4$ -- you can find them by solving a system of 4 equations.
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1In fact, since the matrix is invertible, it follows by the Invertible Matrix Theorem that the columns of the matrix form a basis, hence we are done – 2017-02-27
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0yes, you're right of course, but when solving excercise problems, one often uses the matrix – 2017-02-27
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$\textbf{Hint}$. To show that the system is a base in $\mathbb{R}^4$, it's enough to prove that they are linearly independent. For that, take the determinant of the four vectors and see that it's non zero.
For the second task, remember that the equation of change of basis is
$$X'=P^{-1}X,$$
where $X'$ are the new coordinates, $X$ the old coordinates and $P$ the matrix of change of basis, that it to say, in this case the vectors in columns.