## 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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