Solving Ax = b: the row picture and the column picture
Lesson 1 ended with the column space: it tells us whether Ax = b has a solution. Now we want to find the solution. Two equations can be read in two ways. In the row picture, each equation is a line, like a clue in a treasure hunt that says “the answer is somewhere on me”. The answer is the one spot on both lines, where they cross. In the column picture, the columns of A are two kinds of steps, as in lesson 1, and we ask how many of each step take us to b. Both readings give the same answer.
Lesson 2 · slide 1Rows and columns- Row picture: each equation is a line. The solution is where the lines meet.
- Column picture: the system says x·(column 1) + y·(column 2) = b. Here x and y are plain numbers. The solution gives the right weights.
- Both pictures give the same solution xy. They also fail together: parallel or identical lines always come with dependent columns, and the other way round.
- With 3 or more unknowns the column picture stays easy to think about, so we lean on it.
One system, two readings
Take Professor Strang's first example: x − 2y = 1 and 3x + 2y = 11. As a matrix equation it is Axy = b. Here x and y are numbers; on later slides we call the whole vector xy just x.
The row picture: two clues
Each equation is a line, and each says "the answer is on me". x − 2y = 1 is one line. 3x + 2y = 11 is another. The answer is where they cross: 31. Check: 3 − 2 = 1 and 9 + 2 = 11.
The column picture: a recipe
Now read it by columns, as in lesson 1 (Ax is a combination of the columns):
Now we ask how much of each column makes b. Same answer: x = 3, y = 1.
When the pictures go wrong
- Parallel lines: they never meet, so there is no solution. The columns are dependent and b is off their line.
- The same line twice: infinitely many solutions. b is on the line of the columns.
Why do the two go together? Lines are parallel when one left side is a multiple of the other, like x + y and 2x + 2y. For x + y and 2x + 2y, A = 1122. Both columns are 12, so they lie on one line. The reason is general: if row 2 is k times row 1, every column looks like (a, ka) = a·1k, so all columns are multiples of 1k.
Why the column picture wins later
With three unknowns, each equation is a plane in space (like a line, but in 3D), and three planes are hard to draw. The column picture stays simple: three arrows and one target.
To solve by hand, though, we use rows. That is elimination, on the next slide.
A bigger example: 3 equations, 3 unknowns
By rows, each equation is now a flat plane in space, and the answer is the one point where all three planes meet: xyz = 123. By columns, it is 1 of column 1, 2 of column 2 and 3 of column 3. Engineers solve systems with thousands of unknowns the same way, with a computer doing the arithmetic.
Next: Looking for the crossing point works for two lines, not for a thousand. Slide 2 gives a step-by-step recipe instead: subtract equations to remove the unknowns one at a time. That is elimination.
Examples
Example 1 · One crossing
x + y = 3 and x − y = 1.
Show solutionHide solution
Example 2 · The same line twice
x + 2y = 4 and 2x + 4y = 8.
Show solutionHide solution
Infinitely many solutions, for example 40 and 02.
Example 3 · No crossing at all
x + y = 2 and 2x + 2y = 5.
Show solutionHide solution
Both pictures agree: no solution.
Example 4
Solve 2x + y = 7 and x − y = 2, then check with the column picture.
Show solutionHide solution
Adding the equations: 3x = 9, so x = 3 and y = 1.
Columns: 3·21 + 1·1−1 = 72 ✓
Example 5
What happens with x + y = 1 and x + y = 3? Describe both pictures.
Show solutionHide solution
Rows: two parallel lines, no crossing, no solution.
Columns: both columns are 11. Their combinations stay on the line through 11, and b = 13 is off it.
Questions and comments
Stuck on something, or found a mistake? Ask here and I'll answer. You sign in with GitHub.