Practice question · Multiple choice
Every linear map is fixed by what it does to a basis, and a matrix is just that record. Why does the same transformation have many different matrices?
Hints
- The matrix's columns are the images of the basis vectors. Ask what happens if you pick different basis vectors.
- A rotation is a rotation whatever coordinates you use. Are its entries the same?
Show the answer
D. Because the matrix records the map relative to a chosen basis
Why
The map is geometric and the matrix is a description in a coordinate system, so changing coordinates changes the entries and not the action. That is what similarity means, and it is why diagonalisation is worth doing: you are hunting for the basis in which the description is simplest.
Practise Linear Transformations
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More questions on Linear Transformations
- T is a linear map from three-dimensional space into the plane. Select every statement that MUST be true.
- A linear map on the plane is completely determined once you know what it does to (1,0) and (0,1) - two…
- If T is linear and T(u) = T(v) for two different vectors u and v, then T(u - v) is the zero vector.
- Order the steps of finding the matrix of a linear transformation of the plane.