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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

In this page, you will learn about the bilinear property of vector dot product.



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Dot product is defined between `vec p, vec q in bbb V` and the result `vec p cdot vec q in RR`. This can be considered as a transformation. What type of transformation is this?

  • unary
  • binary
  • binary
  • ternary

The answer is 'binary', as the input is two vectors. 'bi' means two.

Which of the following property is true for Bilinear transformation?

  • `T(x+y, z)=T(x,z)+T(y,z)`
  • `T(ax,z)=aT(x,z)`
  • `T(ax+y, z)= aT(x,z)+T(y,z)`
  • All the above
  • All the above

The answer is 'All the above'.

This property defines linearity for binary operator `T`. That is why it is named Bilinear.

Consider the dot product as a binary transform `T(vec p, vec q) = vec p cdot vec q`. Is dot product bilinear?

  • Yes
  • Yes
  • No

The answer is 'Yes'. The dot product is distributive over vector addition and also satisfies the scalar multiple property.

 •  Dot product is bilinear.

Bilinear Property: For any vector `vec p, vec q, vec r in bbb V` and `lambda in RR`
`(lambda vec p + vec q) cdot vec r = lambda (vec p cdot vec r) + (vec q cdot vec r)`

Solved Exercise Problem:

Given `vec x cdot vec z = 2` and `vec y cdot vec z = 1`, what is `(3 vec x + 2 vec y) cdot vec z`?

  • `5`
  • `6`
  • `8`
  • `8`
  • `9`

The answer is '`8`'
`(3 vec x + 2 vec y) cdot vec z`
`quad = 3 vec x cdot vec z + 2 vec y cdot vec z`

                            
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