Definition: Dot Product. Let
with and , with . The dot product of and is denoted as and is defined by So, dot product in is a map
- The geometric interpretation of the dot product is that it shows orthogonality between vectors in
- When two vectors are orthogonal, the dot product will be zero
- When they are aligned, it will be equal to the product of their magnitudes
- The computation of the dot product is very simple
- simply line up the components, multiply, and sum
- Length relates to dot product in the following way (note that
is another notation for the dot product) . We know that is the length of
Bilinearity
Definition: Bilinear map. A map of
is called bilinear if is linear map in one component when the other component is fixed.
- Essentially, this is saying that the map acts linear for one component at a time
- Meaning that the map is consistent under scalar multiplication and addition when one of the values is held constant
Example: Bilinearity for the dot product in
and Then, is linear (as the fist element is fixed to ) is also linear.
Inner product
Example:
Say instead of , we have This would be similar to dot product, but a bit scaled in one of the axes
- This introduces the idea of the inner product, a concept that is introduced in class but not fully developed
- From Wikipedia: Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors
Remark: Inner product in
. All dot products are a class of inner products. All inner products are bilinear, but not all bilinear maps are inner products.
Remark: There are a class of matrices called positive definite matrices. If
is positive definite, then defines an inner product and after a choice of basis, all inner products are of this above form. Positive definite means that for and only if .