Linear Transformation and Inner Product Problem

Then the standard representation of T is given by the matrix [1,0;0,1].In summary, the conversation discusses the vector space R2 with the standard inner product and the transformation T(v)=AT*v, as well as the use of orthonormal basis to find the matrix representation of T. The solution for part a is easily obtained by expanding T(v)=AT*v, while for part b, the standard representation of T can be found by choosing the base vectors (1,0) and (0,1).
  • #1
i_hate_math
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2

Homework Statement


  1. Consider the vector space R2 with the standard inner product given by ⟨(a, b), (c, d)⟩ = ac + bd. (This is just the dot product.)

    PLEASE SEE THE ATTACHED PHOTO FOR DETAIlS

Homework Equations


T(v)=AT*v

The Attempt at a Solution


I was able to prove part a. I let v=(v1,v2) and w=(w1,w2)
apply T(v)=AT*v and do the expansions easily yields the result

But for part b I am clueless. Please suggest an attempt
 

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  • #2
Hint: the expression 'standard representation of ##T##' means 'representation of ##T## in an orthonormal basis'.
 
  • #3
You can choose any pair of independent vectors u and v to find the matrix elements of the transformation T. Let be these two vectors the base vectors of R2, (1,0) and (0,1).
 

FAQ: Linear Transformation and Inner Product Problem

What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the operations of vector addition and scalar multiplication.

What is an inner product?

An inner product is a mathematical operation that takes two vectors and produces a scalar value. It is often used to measure the angle between two vectors or to determine the length of a vector.

How are linear transformations and inner products related?

Linear transformations and inner products are closely related because inner products can be used to define and analyze linear transformations. Inner products can also be used to determine if a transformation is linear or not.

What is the role of matrix multiplication in linear transformations?

Matrix multiplication is used to represent and perform linear transformations. Each column of the matrix represents the transformation of a basis vector, and the resulting matrix can be used to transform any vector in the original vector space.

How are linear transformations and inner product problems used in real-world applications?

Linear transformations and inner product problems are used in a variety of fields, including physics, engineering, and computer graphics. They are used to model and manipulate real-world systems, such as electronic circuits, mechanical systems, and computer animations.

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