Change of Basis
The same map has different matrices in different bases. A change-of-basis matrix converts coordinates, and similar matrices are the same map seen from two viewpoints.
What a learner can do afterwards
- Convert coordinates between two bases with a transition matrix
- Write the matrix of a map in a new basis by conjugating
- Explain what stays the same about a map when the basis changes
1 · Read
The same vector has different coordinates in different bases, and the transition matrix is the dictionary between the languages. Its columns are the old basis vectors written in the new coordinates. Example: (5, 1) = 3(1, 1) + 2(1, minus 1), so its B-coordinates are (3, 2). Label every vector with its basis.
The same map has different matrices too: A new = P inverse A P, read as translate, apply, translate back. Here P has the new basis vectors as its columns, so P turns new coordinates into old ones. With A = [[1, 2], [0, 3]] and P = [[1, 1], [0, 1]], the new matrix is [[1, 0], [0, 3]], whose (1, 2) entry is 0. Conversion runs through an inverse, so it works in both directions.
To fit a projection, use a basis along its line and across it. Projection onto the line y = x keeps (1, 1) and sends (1, minus 1) to zero. In the basis (1, 1), (1, minus 1), those images have new coordinates (1, 0) and (0, 0), so the matrix is [[1, 0], [0, 0]]: keep the first coordinate, drop the second. In standard coordinates the same map is [[0.5, 0.5], [0.5, 0.5]].
The map stays fixed while only its description changes, so some quantities are immune. The trace, the sum of the diagonal entries, and the determinant belong to the transformation itself. So do eigenvalues: a number lambda is an eigenvalue when A v = lambda v for some nonzero v, which describes what the map does, not how the matrix is written. The diagonal matrix [[3, 0], [0, 8]] has trace 3 + 8 = 11 in every basis, and no basis change moves its determinant, 24.
Most change of basis errors come from untracked labels. Write the basis next to every coordinate vector before you multiply. If a result looks strange, check first whether two languages got mixed in one equation.
Convert coordinates with P, move matrices with P inverse A P, and trust trace, determinant, and eigenvalues to survive the trip.
Reading notes describe open textbooks listed on OER Commons. Videos by Khan Academy. Practice questions are original.
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