4th edition, linear algebra and its application, gilbert strang
exercise 2.1 question 17
- Let P be the plane in R^3 with equation x+y-2z = 4. The origin (0,0,0) is not in P! Find two vectors in P and check that their sum is not in P
Answer at the back –> (4,0,0) is on the plane, (0,4,0) is on the plane but their sum (4,4,0) is not on the plane.
Questions: What does it mean for a vector to be in a plane? Does the end point of the vector being on the plane mean that the vector is in the plane?
This is a central theme in second chapter that strang has used wherein he called the necessity of passing through origin for a subspace as a consequence of closure under addition and closure under scalar multiplication. to quote something – "the distinction between a subset and a subspace is made clear by example. In each case can you add vectors and multiply by scalars, without leaving the space?" and he goes on to demonstrate with a few examples
(Can you ever add two vectors and leave the space, or perform scalar multiplication to the same result?)
Best Answer
To say a vector belongs to a plane is to say that it is "lies" on it. Imagine that you draw an arrow on a sheet of paper. That arrow is a vector, the paper a plane. It lies on the paper.
Now, the neat thing about vector addition and scalar multiplication is that, given an underlying vector space, these operations are closed. You'll soon learn about what a vector space is, but you'll get to linear combinations and spans soon - these are important.