Visualizing a projection onto a plane | Linear Algebra | Khan Academy

TL;DR

This video compares the old and new definitions of projection onto a subspace and shows that they are essentially equivalent.

Transcript

I'm going to do one more video where we compare old and new definitions of a projection. Our old definition of a projection onto some line, l, of the vector, x, is the vector in l, or that's a member of l, such that x minus that vector, minus the projection onto l of x, is orthogonal to l. So the visualization is, if you have your line l like this,... Read More

Key Insights

• 🫥 The old definition of projection onto a line involves finding a vector in the line that results in an orthogonal difference when subtracted from the original vector.
• 😑 The new definition extends the concept of projection to any subspace and requires expressing the original vector as the sum of the projection and a vector in the orthogonal complement of the subspace.
• 🫥 The new definition allows for visualization of projections onto various subspaces, not just lines.
• 🙈 Both definitions of projection are consistent and can be seen as linear transformations.

Questions & Answers

Q: What is the old definition of projection onto a line?

According to the old definition, the projection onto a line is the vector in the line that, when subtracted from the original vector, results in an orthogonal vector to the line.

Q: How does the new definition extend the concept of projection?

The new definition states that the projection onto any subspace is a unique vector in the subspace such that the original vector can be expressed as the sum of the projection and a vector in the orthogonal complement of the subspace.

Q: How can the concept of projection be visualized for a plane?

For a plane as the subspace, the projection of a vector onto the plane is the unique vector in the plane that, when added to a vector in the orthogonal complement of the plane, results in the original vector.

Q: Are the old and new definitions of projection equivalent?

Yes, the video demonstrates that the new definition is essentially equivalent to the old definition for lines, and can be extended to arbitrary subspaces.

Summary & Key Takeaways

• The old definition of projection onto a line states that the projection is a vector in the line that, when subtracted from the original vector, results in an orthogonal vector to the line.

• The new definition expands the concept of projection to any subspace and states that the projection is a unique vector in the subspace such that the original vector can be expressed as the sum of the projection and a vector in the orthogonal complement of the subspace.

• The video provides visualizations of projections onto lines and planes to help understand the concept.