## Hyperplane in $n$-Dimensional Space Through Origin is a Subspace

## Problem 352

A **hyperplane ** in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors

\[\begin{bmatrix}

x_1 \\

x_2 \\

\vdots \\

x_n

\end{bmatrix}\in \R^n\]
satisfying the linear equation of the form

\[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\]
where $a_1, a_2, \dots, a_n$ (at least one of $a_1, a_2, \dots, a_n$ is nonzero) and $b$ are real numbers.

Here at least one of $a_1, a_2, \dots, a_n$ is nonzero.

Consider the hyperplane $P$ in $\R^n$ described by the linear equation

\[a_1x_1+a_2x_2+\cdots+a_nx_n=0,\]
where $a_1, a_2, \dots, a_n$ are some fixed real numbers and not all of these are zero.

(The constant term $b$ is zero.)

Then prove that the hyperplane $P$ is a subspace of $R^{n}$ of dimension $n-1$.

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