Geometrical description of a subset

[dlevanchuk]

Could somebody explain to me please how to figure out a geometric description of a subspace? I understand how to check wether the set of vectors is a subset, but how t ogive them a geometric description??

lets say i have a subset in R3 {x: x3 = 2x1-x2}

why the G.D. is a plane with an equation 2x1-x2-x3 = 0??

or if I have subset {x: aTx = 0}, where a = [1;0; 0] (R3 again), the G.D is a plane yz... huh? how come? :(

 

[mrbohn1]

In the first example you just need to rearrange the set notation so that it reads:
{x: 2x₁-x₂-x₃ = 0}. This is a plane, just as an equation like y-x+1=0 might describe a line in R².

In the second case, I'm not sure what you mean by T, but I'm assuming that by aTx you just mean the vector multiplication of x by the transpose of a. In this case, note that in order to satisfy the equation, x can have non-zero y and z coordinates, but must have a zero x coordinate...ie. the equation defines the plane formed by the y and z-axes.

Note I have used a bold font for vectors...life can get a bit confusing if you use the same notation for vectors and points!

 

[dlevanchuk]

In the first example you just need to rearrange the set notation so that it reads:
{x: 2x₁-x₂-x₃ = 0}. This is a plane, just as an equation like y-x+1=0 might describe a line in R².

In the second case, I'm not sure what you mean by T, but I'm assuming that by aTx you just mean the vector multiplication of x by the transpose of a. In this case, note that in order to satisfy the equation, x can have non-zero y and z coordinates, but must have a zero x coordinate...ie. the equation defines the plane formed by the y and z-axes.

Note I have used a bold font for vectors...life can get a bit confusing if you use the same notation for vectors and points!

Gotcha. So i just have to look at the subset, and rearrange the equation, until it will look familiar (ie plane in 3d or a line in 2d..)

Thanks!