Linear algebra involving dot product and orthongal matrices

• ItsKP
This means that the length of a vector x is the same as the length of its image under R.In summary, The given problem involves showing that the length of a vector x is equal to the length of its image under an orthogonal matrix R. This is because R is an orthogonal matrix, which means that the length of a vector x is the same as its transpose multiplied by the vector y. This information is crucial in solving the problem.
ItsKP

Homework Statement

Given: x (dot) y = x^T * y (where x,y are vectors; dot is dot product; and x^T is x transpose)
and R is an orthogonal nxn matrix, and x,y are elements of R^n

Show ||Rx|| = ||x||

The Attempt at a Solution

I'm not sure what information I am suppose to use to solve this. This is what I have gathered so far:
x(dot)y = x1y1 + x2y2+ ... + xnyn = x^T * y

But how am I suppose to use any of this to solve the problem? Is it involving inner products at all?

ItsKP said:

I'm not sure what information I am suppose to use to solve this.

The crucial information is that R is an orthogonal matrix: R^TR=RR^T=Identity matrix.

1. What is a dot product in linear algebra?

A dot product is a mathematical operation that takes two vectors and returns a single number. It is also known as an inner product or scalar product. In linear algebra, the dot product is used to measure the similarity or perpendicularity of two vectors.

2. How is the dot product calculated?

The dot product is calculated by multiplying the corresponding components of two vectors and then adding the results. For example, if vector A is [a1, a2, a3] and vector B is [b1, b2, b3], the dot product would be a1b1 + a2b2 + a3b3.

3. What is the purpose of orthogonal matrices in linear algebra?

Orthogonal matrices are square matrices whose columns and rows are orthogonal unit vectors. This means that the dot product of any two columns (or rows) is equal to 0, and the norm (length) of each column (or row) is equal to 1. In linear algebra, orthogonal matrices are used for transformations and rotations, and they have many applications in areas such as computer graphics and physics.

4. How do you determine if two vectors are orthogonal?

To determine if two vectors are orthogonal, you can calculate their dot product. If the dot product is equal to 0, then the vectors are orthogonal. Another way to determine orthogonality is to check if the angle between the vectors is 90 degrees.

5. Can two non-orthogonal vectors be made orthogonal?

Yes, two non-orthogonal vectors can be made orthogonal through a process called Gram-Schmidt orthogonalization. This involves finding a vector that is perpendicular to both original vectors and then subtracting its projection from the original vectors. This process can be repeated to create a set of orthogonal vectors.

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