How to find a standard matrix for a transformation?

How could you find a standard matrix for a transformation T : R2 → R3 (a linear transformation) for which T([v1,v2]) = [v1,v2,v3] and T([v3,v4-10) = [v5,v6-10,v7] for a given v1,...,v7? I have been thinking about using a function but do not think this is the most efficient way to solve this question. Could anyone help me out here? Thanks in advance. Walter

[Jan]

Using a function or not is not the question here. It does not matter if you calculate this in a function or directly in the code.

Answer 1

[Roger Stafford]

Assuming the transformation is homogeneous - that is, it leaves the origin fixed - what you have here is six linear equations with six unknown coefficients. Just use standard matlab methods for solving them.

If the transformation is not necessarily homogeneous, then you don’t have enough information for a solution. You would need three instead of the two equalities above.

[Matt J]

And if the transformation is not homogeneous, then strictly speaking it should not be called "linear"...

[Roger Stafford]

Yes, Matt, you are right. This is a difference between linear transformations and linear equations.

[Walter Nap]

Thank you very much for the answer, but the problem here is that I know perfectly fine how to do this by hand (at least, we have learnt to transform the input vectors into elementary vectors [1,0,0], [0,1,0] etc.). But have a problem doing this in computer language. 'Just use the standard matlab methods' is not that standard for me. I just started using the program and have trouble with especially transformations.

Answer 2

[Matt J]

T=[v1,v2,v3;v5,v6-10,v7].'/[v1,v2; v3,v4-10].'

[Walter Nap]

Thank you for your answer, but why do you have to take the transpose of both x and T(x)? Thank you

[Matt J]

So that x and T(x) represent column vectors. Personal choice...