Use basis matrices and distinguish a vector from its coordinate list.
Loading your saved responses…
Try
The same vector v=(5,−1) has coordinates (2,3) in B=((1,1),(1,−1)). The coordinate tuple changes because the reference directions change, not because the vector moves.
Predict
Predict the direction of conversion. Let P_B contain the B vectors as standard columns. Then v=P_B[v]_B. To extract B coordinates from standard coordinates, multiply by P_B⁻¹. Both formulas rely on B being a basis, which makes P_B square and invertible.
Experiment
Reconstruct (5,−1) from 2(1,1)+3(1,−1). Now convert to C=((2,0),(0,1)): the C coordinates are (5/2,−1). Verify P_C⁻¹P_B(2,3) gives that same result. Label coordinate systems on each intermediate tuple.
Understand
Worked operator conversion: suppose the standard matrix is A=diag(2,3). An input in B coordinates first becomes P_B[v]_B, then A acts, then P_B⁻¹ converts the output back. Therefore [T]_B=P_B⁻¹AP_B. Here P_B⁻¹=(1/2)[[1,1],[1,−1]], giving [[5/2,−1/2],[−1/2,5/2]]. Its off-diagonal entries do not mean the underlying map changed.
For different domain and codomain bases, the matrix is P_C⁻¹AP_B. The same-basis operator case is called similarity. It preserves the transformation’s intrinsic eigenvalues, though individual entries can change.
Invalid shortcut: reverse the basis matrices because both are “change of basis.” Their order is determined by the input and output labels; apply the rightmost matrix first.
Apply
Check any conversion by reconstructing the physical or abstract vector. For an operator, test its action on at least one basis vector and interpret the output column in the new basis, not in standard coordinates.
Continue
Depth: a well-chosen basis can simplify repeated applications of a map. Diagonalization seeks an eigenvector basis for this reason. Not every operator admits one, and changing coordinates does not repair a genuinely missing independent direction.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
B=((1,1),(1,−1)). If [v]B=(2,3), enter the standard x coordinate of v.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For the same B and v=(8,2), enter the second B coordinate.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
If PB and PC contain basis vectors as standard columns, [v]C equals:
Changing a basis while retaining the same geometric vector generally:
Derive the matrix of a linear operator with standard matrix A in basis B.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For A=diag(2,3) and B=((1,1),(1,−1)), compute the B-coordinate matrix and verify its first column.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
