## Linear Algebra and Its Applications, Exercise 2.1.6

Exercise 2.1.6. Consider the equation $x+2y+z=6$ that defines a plane $P$ in 3-space. For the parallel plane $P_0$ through the origin find its equation and explain whether $P$ and $P_0$ are subspaces of $\mathbf{R}^3$.

Answer: The origin $(0,0,0)$ must be a solution of the equation for $P_0$. The equation $x+2y+z=0$ satisfies this criterion and its plane $P_0$ is parallel to the original plane $P$.

$P$ is not a subspace because it does not contain the origin, and thus if $v$ is in $P$ then $0 \cdot v$ will not be. $P_0$ is a subspace of $\mathbf{R}^3$ since the sum of any two vectors $u$ and $v$ in $P_0$ is also in $P_0$ as is the scalar multiple $cv$ for any vector $v$ in $P_0$ and any scalar $c$ (including $c=0$).

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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