bernoli123 said:if A [tex]\in[/tex] C nxn,show that (x,Ay)=0 for all x,y [tex]\in[/tex] C[n],
then A=0
(x,Ay) denote standard inner product on C[n]
A standard 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 standard inner product is a specific type of inner product that has certain properties, such as being bilinear and symmetric. Regular inner products may not have these specific properties and can vary depending on the context in which they are used.
Some common properties of a standard inner product include being bilinear (meaning it distributes over addition and scalar multiplication), symmetric (meaning the order of the vectors does not matter), and positive definite (meaning the result is always positive unless the vectors are both zero).
The concept of a standard inner product is used in a variety of fields, including mathematics, physics, engineering, and computer science. It is particularly useful in areas that involve vector spaces, such as linear algebra, functional analysis, and signal processing.
The calculation of a standard inner product involves multiplying the corresponding elements of the two vectors and then summing the products. For example, the standard inner product of two vectors, a and b, in a three-dimensional space would be calculated as a1b1 + a2b2 + a3b3.