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The moment you lost the thread
“The residual is orthogonal to the column space.”
In plain words
Think of a projection as a shadow on a plane. The residual is the part the shadow can’t reach. It points straight off the plane.
Missing a piece?
It connects back to what you learned in week 4.
One moment. Explained in plain words, in course terms, and with the missing context.
Classes › Linear Algebra › Week 6 — Projections and least squares
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Week 6 — Projections and least squares
Recording and transcript
Hide transcript · 61 lines4 words corrected from context
- …so we want the closest point in the column space to b. Closest means the error is as short as it can be.
- The residual is just what the projection leaves behind, which is why it's orthogonal to the column space.
- That gives us the normal equations, and from there the least-squares solution falls straight out.
Advanced · confusion pins
In plain words
Think of the projection as the shadow b casts onto the plane. The residual is the bit of b the shadow can't reach. It points straight off the plane, which is exactly what “orthogonal” means here.Properly
The projection is p = A(AᵀA)⁻¹Aᵀb. The residual b − p satisfies Aᵀ(b − p) = 0, so it is orthogonal to every column of A, and therefore to the whole column space.What you were probably missing
Column space, from week 4: every vector A can produce. You pinned that one too. Without it, “orthogonal to the column space” is a phrase rather than a picture.Key terms
- Projection:
- the closest point to b that A can reach.
- Residual:
- b minus that point; the part A cannot explain.
Check yourself
If b is already in the column space, what is the residual, and why?How it works
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Week 6 — Projections and least squares
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Recording and transcript
Hide transcript · 61 linesPinned at 41:38
- …so we want the closest point in the column space to b. Closest means the error is as short as it can be.
- The residual is just what the projection leaves behind, which is why it's orthogonal to the column space.
- That gives us the normal equations, and from there the least-squares solution falls straight out.
Advanced · confusion pins
It explains that passage three ways.
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Every explanation is built from your own material, in the order your course covered it. It ends with the key terms and one question to check yourself on.
Advanced · confusion pins
In plain words
Think of the projection as the shadow b casts onto the plane. The residual is the bit of b the shadow can't reach. It points straight off the plane, which is exactly what “orthogonal” means here.Properly
The projection is p = A(AᵀA)⁻¹Aᵀb. The residual b − p satisfies Aᵀ(b − p) = 0, so it is orthogonal to every column of A, and therefore to the whole column space.What you were probably missing
Column space, from week 4: every vector A can produce. You pinned that one too. Without it, “orthogonal to the column space” is a phrase rather than a picture.Key terms
- Projection:
- the closest point to b that A can reach.
- Residual:
- b minus that point; the part A cannot explain.
Check yourself
If b is already in the column space, what is the residual, and why?Study from the moments you marked.
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Why is the residual orthogonal? I don't see where that comes from.
Your slides define the projection as the closest point in the column space. If the residual had any part along the column space, you could move the projection that way and get closer, so it can't. That's the whole argument on slide 14.
Lecture 12, slide 14 · Strang §4.2Everything in the app
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