The essence of linear algebra, the source video www.bilibili.com/video/BV1ys…

Dot product and duality

What is the dot product?

• Calculation of level
  
  If you take the dot product of two vectors with the same dimension, you pair the corresponding coordinates, take the product of each pair of coordinates and add them.
  
  
• The geometric angles
  
  This is the dot product in the usual sense, and we’ll see what it means in the set sense, is to project one vector in the direction of another vector, times the length of the other vector, and you get the dot product.
  
  

Why can two dot product projections be in different directions? We can use symmetry to illustrate this.

Why are these two approaches related? Where is their connection?

For a 1 by 2 vector, we can translate it to a one-dimensional number line.

The first one of the vectors is pretty much the transformed oneiThe second one is equivalent to the converted onejThink of it as a special linear transformation.

Any time we see a linear transformation, its output space is a one-dimensional number line, no matter what form it takes, there will be a unique vector v that corresponds to it, and in that sense, applying a linear transformation is the same as taking the dot product of vector V.

Duality = natural and unexpected correspondence.