Introduction to projections | Matrix transformations | Linear Algebra | Khan Academy | Summary and Q&A

TL;DR

Projections are a way to find the shadow or amount of a vector that lies in a specific direction or line.

Key Insights

• 🫥 A line through the origin in any dimension can be represented as the set of scalar multiples of a vector.
• 🫥 The projection of a vector onto a line can be visualized as its shadow or the amount of the vector that lies in the direction of the line.
• 🫥 The projection can be defined as a vector in the line that is orthogonal to the difference between the original vector and the projection.

Transcript

Let's say I have a line that goes through the origin. I'll draw it in R2, but this can be extended to an arbitrary Rn. Let me draw my axes. Those are my axes right there, not perfectly drawn, but you get the idea. Let me draw a line that goes through the origin here. So that is my line there. And we know that a line in any Rn-- we're doing it in R2... Read More

Questions & Answers

Q: How can a line in any dimension be defined?

A line in any dimension can be represented as the set of all possible scalar multiples of a vector.

Q: How can the projection of a vector onto a line be described?

The projection can be thought of as the shadow of the vector or the amount of the vector that lies in the direction of the line.

Q: How can the projection be mathematically defined?

The projection can be defined as a vector in the line that is orthogonal to the difference between the original vector and the projection.

Q: How can the projection be calculated?

The projection can be calculated using the dot product between the original vector and the defining vector of the line, divided by the dot product of the defining vector with itself, multiplied by the defining vector of the line.

Summary & Key Takeaways

• A line in any dimension can be represented as a set of scalar multiples of a vector.

• The projection of a vector onto a line can be thought of as its shadow or the amount of the vector that lies in the direction of the line.

• The projection can be mathematically defined as a vector in the line that is orthogonal to the difference between the original vector and the projection.

• The projection can be calculated using the dot product and vector scalar multiplication.