The transition matrix P_{C←B} that converts B-coordinates to C-coordinates is constructed by:
AWriting the C-basis vectors in B-coordinates as the columns
BWriting the B-basis vectors in C-coordinates as the columns
CWriting the C-basis vectors in standard coordinates as the columns
DWriting the B-basis vectors in standard coordinates as the rows
P_{C←B} converts B-coordinates to C-coordinates. Its columns are the B-basis vectors expressed in C-coordinates — you ask 'what is this B-basis vector, described in C's language?' and that answer becomes a column. This ensures [P_{C←B}][v]_B = [v]_C.
Question 2 Multiple Choice
A linear transformation T has matrix A in standard coordinates. In basis B (with B-matrix P whose columns are the B-basis vectors), the same transformation is represented as:
APAP⁻¹
BP⁻¹A
CP⁻¹AP
DPᵀAP
The sandwich P⁻¹AP encodes: P converts from B-coordinates to standard, A applies the transformation in standard coordinates, P⁻¹ converts back from standard to B-coordinates. Matrices related this way are called similar and represent the same transformation from different coordinate perspectives.
Question 3 True / False
Two similar matrices A and P⁻¹AP always represent the same linear transformation, just described in different coordinate systems.
TTrue
FFalse
Answer: True
Similarity transformation is the mathematical expression of 'same transformation, different coordinate description.' P⁻¹AP encodes the same geometric mapping as A — the same action on vectors — using the coordinate language of basis B rather than the standard basis.
Question 4 True / False
The change-of-basis matrix P_{C←B} and its inverse P_{B←C} are transposes of each other.
TTrue
FFalse
Answer: False
They are inverses: P_{B←C} = (P_{C←B})⁻¹. Transposes and inverses coincide only for orthogonal matrices (where basis vectors are orthonormal). In general, converting B→C and then C→B composes to the identity, which is an inverse relationship, not a transpose relationship.
Question 5 Short Answer
Why does choosing the eigenvector basis for a linear transformation make it so much easier to analyze the transformation?
Think about your answer, then reveal below.
Model answer: In the eigenvector basis, P⁻¹AP yields a diagonal matrix — each eigenvector is simply scaled by its eigenvalue, with no mixing between components. A diagonal matrix is trivial to raise to powers, invert, or compose: just apply the operation to each diagonal entry independently. What looked like a complicated interaction in standard coordinates becomes independent scaling along each eigendirection.
This is the core payoff of change of basis. The transformation hasn't changed — but the right coordinate description reveals its structure. Choosing the basis that matches the transformation's natural directions converts a hard problem into a transparent one.