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dimensionless
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Is there any difference between an inner product and a dot product?
Just expanding on what arildno said, one specific example is the Frobenius inner product of two matrices. It is defined by:dimensionless said:What would be an example of an inner product that it not a dot product?
For continuous functions from [a,b] to R, you can define an inner product <.,.> as:dimensionless said:What would be an example of an inner product that it not a dot product?
(a, b) . (c, d) = ac + 2bdWhat would be an example of an inner product that it not a dot product?
Hurkyl said:(a, b) . (c, d) = ac + 2bd
(a, b) . (c, d) = (a+b)(c+d) + (a-b)(c-d)
(a) . (b) = 2ab
arildno said:General terms:
Inner/Outer products
No -- they are definitions of three different inner products. The first two are on R², and the third is on R.Are these postulates?
An inner product is a mathematical operation that takes two vectors and returns a scalar value. It is defined as the sum of the products of the corresponding elements of the two vectors. A dot product, on the other hand, is a specific type of inner product that is only defined for vectors in Euclidean space (where the length and angle between vectors are well-defined). It is calculated by taking the sum of the products of the corresponding elements of the two vectors, but also taking into account the angle between them.
No, they cannot. While a dot product is a type of inner product, not all inner products are dot products. Inner products can be defined for other types of vector spaces, such as complex vector spaces, while dot products are specific to Euclidean spaces.
Inner products and dot products have many applications in mathematics, physics, and engineering. For example, they are used in signal processing, computer graphics, and machine learning algorithms. They are also used in physics to calculate work and power, and in quantum mechanics to calculate probabilities and expectation values.
No, they are not. The commutative property states that the order of operands does not affect the result of an operation. However, the order of vectors does affect the result of an inner product or dot product, as the angle between vectors is taken into account.
In both inner products and dot products, two vectors are considered orthogonal if their product is equal to zero. However, in inner products, orthogonality is a result of the definition of the operation, while in dot products, orthogonality is a property of Euclidean space.