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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.
mathsProperties of Vector ArithmeticsProperties of Dot Product

### Bilinear Property

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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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