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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