Linear Algebra: Linearity of inner product

In summary: If we assume you are speaking of C^n with <x,y> = \sum_{i=1}^n \bar{x_i} y_i, it is definitely linear in both x and y. Your "equation"<\text{conjugate}(y),\text{conjugate}(x)> = <x,y> is false, if by "conjugate" you mean "complex conjugate". It is not even true in one dimension, with <x,y> = \bar{y}x.
  • #1
teddyayalew
36
0

Homework Statement


An inner product is linear in both components.

Homework Equations


<x,y> = <conjugate(y),conjugate(x)>
<x+y,z> = <x,z> +<y,z>


The Attempt at a Solution



I thought it was true . It is obvious that it is linear for the first component by definition
Attempt to show it is for second component:

<x,y+z> = <conjugate(y+z),conjugate(x)>
=<conjugate(y),conjugate(x)> + <conjugate(z),conjugate(x)>
= <x,y> + <x,y>


But the answer is false. I am having trouble understanding why it is not linear in both components. The answer key says that the second component is conjugate- linear.
 
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  • #2
teddyayalew said:

Homework Statement


An inner product is linear in both components.

Homework Equations


<x,y> = <conjugate(y),conjugate(x)>
<x+y,z> = <x,z> +<y,z>


The Attempt at a Solution



I thought it was true . It is obvious that it is linear for the first component by definition
Attempt to show it is for second component:

<x,y+z> = <conjugate(y+z),conjugate(x)>
=<conjugate(y),conjugate(x)> + <conjugate(z),conjugate(x)>
= <x,y> + <x,y>


But the answer is false. I am having trouble understanding why it is not linear in both components. The answer key says that the second component is conjugate- linear.

What is the vector space you are working in? What is the relevant definition of inner product?

RGV
 
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  • #3
It doesn't say which space. It was just a true or false statement, but I know that the entries of the vectors can be from the complex space. And it doesn't tell me which definition of the inner product but I believe it is the standard inner product.
 
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  • #4
teddyayalew said:
It doesn't say which space. It was just a true or false statement, but I know that the entries of the vectors can be from the complex space. And it doesn't tell me which definition of the inner product but I believe it is the standard inner product.

In your OP YOU said "It is obvious that it is linear for the first component by definition", implying that you had some definition in mind--otherwise, why use the word 'definition'? So, are you now saying that is not true?

If we assume you are speaking of [itex]C^n[/itex] with [itex] <x,y> = \sum_{i=1}^n \bar{x_i} y_i, [/itex] it is definitely linear in both x and y. Your "equation"
[tex] <\text{conjugate}(y),\text{conjugate}(x)> = <x,y>[/tex] is false, if by "conjugate" you mean "complex conjugate". It is not even true in one dimension, with [itex] <x,y> = \bar{y}x.[/itex]

RGV
 

1. What is the definition of linearity of inner product in linear algebra?

The linearity of inner product states that the inner product of two vectors is a linear function of each vector, meaning that it follows the properties of linearity such as additivity and homogeneity. In other words, the inner product of two vectors is equal to the sum of the inner products of each vector with a third vector, and it is also equal to a constant times the inner product of the first vector with a scaled version of the second vector.

2. How is linearity of inner product related to vector spaces?

The linearity of inner product is an essential property in vector spaces. It ensures that the inner product is a well-defined operation in a vector space, and it also allows for the use of important concepts such as orthogonality and projections. Without linearity of inner product, these concepts would not hold true in vector spaces.

3. What are the key properties of linearity of inner product?

The key properties of linearity of inner product are additivity and homogeneity. Additivity states that the inner product of two vectors is equal to the sum of the inner products of each vector with a third vector. Homogeneity states that the inner product of a vector with a scalar multiple of another vector is equal to the scalar multiple of the inner product of the first vector with the second vector.

4. Can the linearity of inner product be extended to matrices?

Yes, the linearity of inner product can be extended to matrices. This is known as the linearity of matrix inner product. It states that the inner product of two matrices is equal to the sum of the element-wise inner products of the matrices. It also follows the properties of linearity, such as additivity and homogeneity.

5. How is the linearity of inner product used in practical applications?

The linearity of inner product is a fundamental concept in linear algebra, and it is used in various practical applications such as data analysis, image processing, and machine learning. It allows for the manipulation and transformation of vectors and matrices, making it a powerful tool in solving real-world problems.

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