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Let $\alpha_1,...,\alpha_n,\beta\in$ field $\mathbb R.$ Which properties should the given scalars have so that $M=\{(x_1,...x_n)\in \mathbb R^n: \displaystyle\sum_{i=1}^{n} \alpha_ix_i=\beta\}$

would be the subspace of $\mathbb R^n$?

Find the dimension of $M$.

I know that the subspace must include all the linear combinations of its elements (vectors). First, I checked if it is closed under adition. I planned to check for the multiplication in the next step, but it really got complicated. I used some vectors $x$ and $y$ and scalars $\alpha$ and $\gamma$ for each. After that I used $\lambda_i$ for their sum and got (after the coordinate adition), since I wrote the first coordinate in respect to the constraint: $\lambda_1\gamma_1\alpha_i +\lambda_1\alpha_1\gamma_i=\alpha_1\gamma_1\lambda_1$. May I ask for suggestion?

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    $\begingroup$ $M$ is a subspace iff $\beta=0$ and the dimension is $n-1$ or $n$ according as $\alpha_i$ are all $0$ or one of then is non-zero. $\endgroup$ – Kavi Rama Murthy Nov 7 '19 at 12:40
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A subspace must always contain the zero vector; this means (after a trivial computation) that $\beta=0$ is a necessary condition. You may check that the condition is also sufficient, as the solution of a linear homogeneous equation (or of a system of such equations) is always a subspace.

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  • $\begingroup$ That was one idea, but after imagining the horrendous combinations" I wasn't sure if I was sure" . Thank you very much. $\endgroup$ – Praskovya2.718281828 Nov 7 '19 at 12:46

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