Lesson 10 of 23 · 2:41
Swap operands by transposing a CM
Explain a transpose with its operand order and labelled axes.
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In this lesson
- 0:00 · Does X implying Y mean that Y implies X?
- 0:23 · Locate the only failure
- 0:39 · Transpose: exchange row and column indices
- 0:56 · Keep the measurement identity visible
- 1:13 · Calculate the reversed example
- 1:29 · A symmetric operator stays the same
- 1:46 · Your turn: predict the transposed operator
- 2:06 · Answer: return to the original
- 2:21 · Separate transformation from relabelling
Complete transcript
This transcript includes the answers to prediction prompts.
Show transcript
Does X implying Y mean that Y implies X? · 0:00
Does X implying Y mean that Y implies X? Not in general. We will reverse the operands and learn exactly which matrix entries move under a transpose. Keep the true-first convention: the first row and first column mean one. The axis labels will explain which question each matrix answers.
Locate the only failure · 0:23
Implication fails only when its left input is one and its right input is zero. That is the upper-right cell. For X equal to one and Y equal to zero, X implies Y is false, while Y implies X is true.
Transpose: exchange row and column indices · 0:39
A transpose exchanges the row and column indices of every entry. Diagonal entries stay where they are. The upper-right zero moves to the lower-left position. We have now made the matrix of reverse implication under the original X-row and Y-column labels.
Keep the measurement identity visible · 0:56
There are two ways to ask the reversed question. We can put Y on the left of the original implication matrix and X on its right. Or keep X on the left and Y on the right, while transposing the matrix. Both ask whether Y implies X.
Calculate the reversed example · 1:13
Use X equal to one and Y equal to zero again. With X on the left of the reverse-implication matrix, the bra selects its first row, one, one. Ket zero selects the second entry. The result is one, agreeing with zero implies one.
A symmetric operator stays the same · 1:29
Exclusive-or is different: its matrix equals its transpose. The two off-diagonal entries are both one. This tells us that swapping the operands preserves the exclusive-or rule for every input pair. Symmetry is visible in the numeric matrix.
Your turn: predict the transposed operator · 1:46
Transpose the displayed implication matrix once more. Track the zero from its current position, then name the bracketed operator you obtain and explain why every entry returns to its original coordinate.
Prediction pause: 8 seconds.
Answer: return to the original · 2:06
The zero moves from lower left back to upper right, so two transposes recover the implication matrix. Every entry returns to its starting coordinates. This also explains why reversing the input order twice recovers the original question.
Separate transformation from relabelling · 2:21
A matrix transformation has to travel with a statement about the operands and axes. Moving entries under fixed axes can change the represented rule. Moving entries and relabelling the inputs together can preserve a measurement. Next we distinguish input negation from output negation.