Transformations and basis vectors

Question

I've been reading this very nice tutorial on OpenGL, and I encountered a statement which I can't wrap my head around. In Chapter 6, it states:

Transformation from one space to another ultimately means this: taking the basis vectors and origin point from the original coordinate system and re-expressing them relative to the destination coordinate system. The transformation matrix from one space to another contains the basis vectors and origin of the original coordinate system, but the values of those basis vectors and origin are relative to the destination coordinate system.

This is reaffirmed in Chapter 7:

... And this makes sense; a transformation matrix contains the basis vector and origin of the source space, as expressed in the destination coordinate system.

... But isn't it the other way around?

Take a look at this example from the tutorial:

The transformation Matrix to transform from the space on the left to the space on the right is

[1   0   0   1  ]
[0   1   0  -1.5]
[0   0   1   0  ]
[0   0   0   1  ]

The last column of the transformation matrix is the origin of the destination space, expressed relative to the source space.

What am I missing?

Note The above transformation matrix is not from the notes, so feel free to correct it.

Answer 1

The transformation Matrix to transform from the space on the left to the space on the right is

[1   0   0   1  ]
[0   1   0  -1.5]
[0   0   1   0  ]
[0   0   0   1  ]

Well this is not correct, the matrix you have shown is actually to transform any point from the space on the right to the space on the left. The correct matrix to transform from the space on the left to the right is actually

[1   0   0  -1  ]
[0   1   0  1.5 ]
[0   0   1   0  ]
[0   0   0   1  ]

Think about it like this. The [0,0] point on the left figure is actually [-1,1.5] relative to the new space (right figure), remember when you change a basis it's easier to think about it as actually transforming basis not the points, meaning you're actually representing the same points relative to a different basis.

Even though you can think of it the other way around, as of actually taking the inverse transform of the points, it's actually easier to think about it as transforming the basis, either way pick a convention and stick with it. You can also check my other answer, for better explanation of spaces.