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What is the difference between these two objects?

In 2D? In 3D? In 4D?

It would be great if anyone can give me some examples distinguish the two concepts and pictures.

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    $\begingroup$ It's gonna be hard to find a picture of a plane in 4d! $\endgroup$ – qbert Aug 28 '16 at 5:39
  • $\begingroup$ 3d and 2d also OK. $\endgroup$ – math101 Aug 28 '16 at 5:40
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In general, a hyperplane in $\Bbb R^n$ is an $(n-1)$-dimensional subspace of $\Bbb R^n$. So, in the case of $\mathbb R^4$, you may think of a hyperplane as a rotated version of our three-dimensional space $\mathbb R^3$. In $\Bbb R^3$, a hyperplane is a two-dimensional plane, and in $\Bbb R^2$, a hyperplane is a one-dimensional line.

To answer your question about the difference between a plane and a hyperplane, a plane and a hyperplane are the same thing in $\Bbb R^3$. We use the term hyperplane to speak of $(\dim V - 1)$-dimensional subspaces of a $(\dim V)$-dimensional vector space $V$. In the case $n = 3$, geometric planes may be thought of as the classic span of $2 = 3-1$ vectors in $\Bbb R^3$. (The number $1$ is often referred to as the "codimension" of the plane.)

In $\Bbb R^n$, an example hyperplane is defined by the equation $$ a_1x_1 + \dotsb + a_nx_n = 0, $$ where $(a_1,\dots,a_n)$ is not the zero vector. More explicitly, the hyperplane in the discussion is the kernel of the linear map $\Bbb R^n\to\Bbb R$ defined by $$ (x_1,\dots,x_n)\mapsto a_1x_1 + \dotsb + a_nx_n. $$ Since $(a_1,\dots,a_n)$ is not the zero vector, the image of this linear map is a $1$-dimensional subspace of $\Bbb R$ (i.e. is equal to $\mathbb R$), so the rank-nullity theorem tells us that the kernel of this linear map is an $(n-1)$-dimensional subspace of $\Bbb R^n$, i.e., a hyperplane in $\mathbb R^n$. In the case $n = 3$, this is usually how we define a plane orthogonal to the vector $(a_1,a_2,a_3)$, as the set of vectors $(x_1,x_2,x_3)$ satisfying $$ a_1x_1 + a_2x_2 + a_3x_3 = 0. $$ In the sense that geometric planes and kernels of nonzero linear functions $\Bbb R^3\to\Bbb R$ coincide, hyperplanes in $n$-space are the appropriate generalization of geometric planes in $3$-space to arbitrary dimensions.

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