Lesson 6 of 23 · 2:58
Two logical matrices become a 4×4
Expand a tensor product and identify its full row and column components.
Keyboard: focus the player, then Space or K to play/pause and arrow keys to seek. Tab moves through the controls. Selectable captions replace the video's caption strip with the same timed text below.
In this lesson
- 0:00 · How do two small matrices become a larger one?
- 0:22 · Read the two equalities
- 0:38 · Each entry expands into a block
- 0:54 · Inspect the first block
- 1:09 · Fill the remaining blocks
- 1:29 · Understand the row labels
- 1:44 · Trace one expression across the grid
- 2:01 · Your turn: the final expression
- 2:25 · Answer: all four are negated
- 2:41 · The scaffold is ready
Complete transcript
This transcript includes the answers to prediction prompts.
Show transcript
How do two small matrices become a larger one? · 0:00
How do two small matrices become a larger one? Suppose one complete logical matrix uses W and X, and another uses Y and Z. A tensor product keeps every combination of their entries. We will build the entire four-by-four grid one block at a time, then use its row and column components to explain a selected cell.
Read the two equalities · 0:22
M sub W X is the whole matrix on the left of the tensor symbol. M sub Y Z is the whole matrix on the right. The equals signs connect each name to its four expressions. W X inside the first matrix is only one entry.
Each entry expands into a block · 0:38
For an ordinary block expansion, take each entry of the first matrix and combine it with every entry of the second. There are two block rows and two block columns. Each block has two rows and two columns, giving four of each overall.
Inspect the first block · 0:54
Begin with the first block. Every expression there includes W and X. The second matrix contributes Y and Z, Y and not Z, not Y and Z, or neither. These four combinations occupy a square, with two entries in each row.
Fill the remaining blocks · 1:09
The next block uses W and not X. The two lower blocks use not W. In each case the second matrix contributes all four of its expressions. Now the complete grid contains sixteen expressions, covering all combinations of the four variables.
Understand the row labels · 1:29
Why do the rows use W and Y? Each row combines a row choice from the first matrix with a row choice from the second. Each column similarly combines X and Z. The row components and column components are written separately above the grid.
Trace one expression across the grid · 1:44
Follow row W and not Y, then column not X and Z. The intersection contains W, not X, not Y, and Z, joined by AND. The axis components tell you every factor in that expression without memorising the grid.
Your turn: the final expression · 2:01
Find the last row and the last column. Which of the four variables are negated at that intersection? Read the row components first, then the column components. You can pause here if you want more time.
Prediction pause: 10 seconds.
Answer: all four are negated · 2:25
All four variables are negated. The row contributes not W and not Y. The column contributes not X and not Z. Joined together, they give the final expression shown. This is one entry of the full tensor, not the tensor itself.
The scaffold is ready · 2:41
This base tensor organises conjunction expressions. To construct a more interesting compound rule, we must supply the appropriate logical matrices and combine them according to that rule. The next lesson does that for exclusive-or followed by implication.