Subspace Y/N  |  Restriction equations     |  English sentence restriction
===========================================================================

Yes           |  <x,y,z> with z=0          |  xy-plane in R^3

===========================================================================

No            |  x >= 0, y >= 0            |  Quadrant I in the plane

===========================================================================

No            | ?                          |  All points in R^3 with
              |                            |  x,y,z integers

===========================================================================

No            | x+y+z=1                    |  The plane in $R^3 with
              |                            |  normal i+j+k through (0,0,1)

===========================================================================

Yes           | x+y+z=0                    |  The plane in $R^3 with
              |                            |  normal i+j+k through (0,0,0)

===========================================================================

No            | x+y+z=1                    |  The plane in $R^3 with
              |                            |  normal i+j+k through (0,0,1)

===========================================================================

Yes           | Au=0                       |  Linear homogeneous algebraic
              |                            |  equations, solution set     
              |                            |  V=R^n

===========================================================================

No            | Au=b with b not zero       | Linear non-homogeneous algebraic
              |                            | equations, solution set      .
              |                            | V=R^n

===========================================================================

No, not as    | ? no equations             | position (x,y,z) of a slot
defined       |                            | racer on a track atop a table.
              |                            | V=R^3

===========================================================================

Yes           | f(x+2)=f(x)                | All 2-periodic functions
              |                            | V=all functions on
              |                            | -infinity to infinity.

===========================================================================

Yes           | y'' + y = 0 is the         | All solutions of the pendulum
              | restriction equation       | differential equation. V=same
              |                            | vector space above.

===========================================================================

Yes           | color(pixel)=greyscale     | All digital photos 4096x2048
              | or                         | with monchrome processing.
              | B(photo matrix)=zero       | V=all 4096x2048 matrices
              | where B isolates colors    |
===========================================================================