Question

Active and passive transformation linear algebra

Answer 1

In an active transformation, given a basis, we start from a vector and we find a new vector in the same basis.In a passive transformation we have a vector expressed in a basis and we express it in a new basis.

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The figure illustrate the action of a matrix A as an active transformation and of A−1 as the corresponding passive transformation.

Here we have:

A=[1−224]A−1=18[42−21]

The matrix A acts on a vector x that in the standard basis S (represented in black) has components x=[3,2]TS, and, as active transformation, gives the vector x′=Ax=[7,2]TS.

Note that in the new basis B that has as basis vectors the columns of A (represented in blue) this vector has components x′=[3,2]TB.

The inverse matrix A−1 represents the passive transformation that gives the components of the vector x in the new basis B:

A−1x=18[42−21][32]=[11]