The column space: every place Ax can reach
Slide 3 gave one output Ax for one x. Now collect all of them. Each column of A is a kind of step you're allowed to take, say one step north-east and one step north-west. You can take as many of each as you like, forwards or backwards. The column space is every spot you can reach that way. If the two steps go in different directions, you can get anywhere. If they go the same way, you are stuck walking back and forth along one line. Ax = b has a solution (an x that makes Ax equal to b) exactly when b is a spot you can reach.
Lesson 1 · slide 4Column space- The column space C(A) contains every combination of the columns of A, that is, every vector Ax.
- Ax = b has a solution exactly when b is in C(A).
- In the plane, C(A) is the whole plane, a line through the origin, or just the origin.
- Columns are independent when no column is a combination of the others.
- The rank counts how many independent directions C(A) has: 2 for the plane, 1 for a line, 0 for the origin.
All the outputs at once
Try every possible x and mark each Ax on the plane. All those points form the column space, C(A). Every Ax is a combination of the columns, hence the name.
A quick picture: two kinds of steps
Each column is one kind of step. x says how many of each to take (negative means walk backwards). Steps in different directions can take you anywhere: C(A) is the plane. Steps in the same direction only take you forward or back along one line: C(A) is a line.
Three possibilities in the plane
- The whole plane: the columns point different ways. They are independent.
- A line: one column is a multiple of the other. They are dependent.
- Just the origin: both columns are zero.
The rank names the case: 2, 1 or 0.
A closer look at the dependent case
Also, x = 30 and x = 11 both give 36. With dependent columns, different inputs can give the same output. Lesson 4 comes back to this extra freedom.
Two questions that sound alike
- Can we solve Ax = b for this one b? Yes, if b is in C(A).
- Can we solve Ax = b for every b? Only if C(A) is the whole plane.
For two columns there is a quick test: is one column a multiple of the other? If yes, C(A) is only a line (or just the origin). If no, C(A) is the whole plane. Lesson 3 turns this test into a single number, the determinant.
A bigger example: columns in 3D
Take two columns with 3 numbers each. Mixing them gives
The third number is always the first plus the second. So you can't reach all of space, only a flat sheet through the origin. b = 123 is on the sheet (take x = 12). b = 120 is not, so Ax = b has no solution for it.
Add a third column that sticks out of the sheet, like 001 (now x needs three numbers), and now you can reach every point in space. A third column that lies on the sheet, like 112, adds nothing new. The same holds in any size: the column space can be all of space, or a smaller flat piece of it.
Next: The column space tells us whether Ax = b has a solution. Lesson 2 shows how to find it: first by reading the equations two ways, then by elimination. The recap comes first.
Examples
Example 1 · A line, and who is on it
A = 2412. Column 2 = 2·column 1, so C(A) is the line through 21.
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Example 2 · The whole plane
A = 111−1. The columns 11 and 1−1 are not multiples of each other, so C(A) is the whole plane. Solve Ax = 42:
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Example 3 · Is b in the column space?
A = 1326, b = 25.
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So b is not in C(A), and Ax = b has no solution. Changing b to 24 would make it solvable.
Example 4
Describe C(A) for A = 3612. What is the rank?
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Column 2 = 2·column 1, so C(A) is the line through 31. Rank 1.
Example 5
What is C(A) when A is the zero matrix (all entries 0)? And what is C(I), where I = 1001 is the identity matrix (it leaves every vector where it is: Ix = x)?
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Zero matrix: only the origin, rank 0. Identity: the whole plane, rank 2, since Ix = x reaches everything.
Questions and comments
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