Inner Product and Linear Transformation

Then you can use linearity to show how it acts on any vector.In summary, the conversation discusses the existence of a vector u in a finite-dimensional real inner product space V such that a linear map L can be expressed as L(x) = <x,u> for all x in V. The solution involves using an orthonormal basis for V and examining how the linear functional L acts on individual basis elements, using linearity to extend it to any vector in V. This may be related to the Reisz-representation theorem.
  • #1
J-Wang
4
0

Homework Statement


Let V be a finite-dimensional real inner product space with inner product < , >.

Let L:V->R be a linear map.

Show that there exists a vector u in V such that L(x) = <x,u> for all x in V.


2. The attempt at a solution
It seems really simple but I just can't phrase it properly.

I know since L is linear, I can express as L(x+y) = L(x) + L(y)

I tried to make x and u be expressed as combination of orthogonal basis, and try to work in the fact that inner product will result in ZERO since they are orthogonal.

Any help would be greatly appreciated.
 
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  • #2
Since V is a finite dimensional vector space, there exist an orthonormal basis [itex]\{v_i\}[/itex] for i from 1 to n. Let [itex]a_i= L(v_i)[/itex]. What can you say about <x, v> with [itex]v= \sum_{i=1}^n a_iv_i[/itex]?
 
  • #3
Does it has something to do with the Projection of x on the basis? :D
 
  • #4
Are you trying to prove the Reisz-representation theorem?

I would see how this linear functional acted on individual basis elements.
 

FAQ: Inner Product and Linear Transformation

What is an inner product?

An inner product is a mathematical operation that takes two vectors and produces a scalar. It is often used to measure the angle between two vectors or to calculate the length of a vector. In other words, it is a way to quantify the similarity or difference between two vectors.

How is an inner product different from a dot product?

An inner product is a more general concept than a dot product. While a dot product is a specific type of inner product that is defined for vectors in Euclidean space, an inner product can be defined for vectors in other spaces as well. Additionally, an inner product can have different properties and can be defined using different mathematical operations, while a dot product always uses the multiplication and addition of vector components.

What is a linear transformation?

A linear transformation is a function that maps one vector space to another in a way that preserves the operations of vector addition and scalar multiplication. In other words, the output of a linear transformation is a linear combination of the input vectors. Linear transformations are often used in mathematics and physics to describe how objects and systems change or behave.

How are inner products and linear transformations related?

Inner products and linear transformations are related in that some types of linear transformations can be represented using inner products. For example, in the case of dot products, the dot product of two vectors is equal to the inner product of one vector and the linear transformation of the other vector. Additionally, inner products can be used to define and analyze properties of linear transformations, such as orthogonality and unitarity.

What are some real-world applications of inner products and linear transformations?

Inner products and linear transformations have many real-world applications in fields such as physics, engineering, and computer science. Examples include using inner products to calculate work and energy in physics, using linear transformations to compress and manipulate images in computer graphics, and using both concepts in data analysis and machine learning to find patterns and make predictions.

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