Practice question · Put in order
Order the steps of finding the matrix of a linear transformation of the plane.
- Apply T to the second standard basis vector (0, 1)
- Write the result as the first column of the matrix
- Apply T to the first standard basis vector (1, 0)
- Check by computing the matrix times a test vector and comparing with T of that vector
- Write that result as the second column
Hints
- A linear map is determined by what it does to a basis, so start there.
- The images of the basis vectors become COLUMNS, in the order of the basis.
Show the answer
- Apply T to the first standard basis vector (1, 0)
- Write the result as the first column of the matrix
- Apply T to the second standard basis vector (0, 1)
- Write that result as the second column
- Check by computing the matrix times a test vector and comparing with T of that vector
Why
Because every vector is a unique combination of the basis vectors and T preserves combinations, the images of the basis vectors pin down T everywhere. Placing them as columns gives a matrix A with T(v) = Av, a matrix simply IS a linear map written in coordinates.
Practise Linear Transformations
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