Understanding Vector Dot Product and Gaussian vs. Gauss-Jordan Elimination

[Joza]

Oh guys, I'm asking for explanations here, a little lesson.

If something could explain vector dot product (including it's algebraic method) that would be great.

And another thing, what is the difference between Gaussian and Gauss-Jordan elimination?

 

[cristo]

Oh guys, I'm asking for explanations here, a little lesson.

If something could explain vector dot product (including it's algebraic method) that would be great.

What do you mean by explain the dot product? What specifically don't you understand?

And another thing, what is the difference between Gaussian and Gauss-Jordan elimination?

See here: http://en.wikipedia.org/wiki/Gauss-Jordan_elimination

 

[tacman]

Sure, I would be happy to provide some explanations on vector dot product and Gaussian vs. Gauss-Jordan elimination.

Vector dot product, also known as the scalar product, is a mathematical operation that takes two vectors and produces a scalar quantity. This operation is denoted by a dot (·) between the two vectors. It is calculated by multiplying the corresponding components of the two vectors and then adding them together. For example, if we have two vectors A and B, the dot product would be A·B = (A1*B1) + (A2*B2) + (A3*B3). In other words, it is the sum of the products of the corresponding components of the two vectors. Geometrically, the dot product can be thought of as the projection of one vector onto the other.

Now, onto Gaussian and Gauss-Jordan elimination. These are two methods used in linear algebra to solve systems of linear equations. Both methods involve transforming the system of equations into an equivalent system with simpler equations that are easier to solve. The main difference between the two methods is that Gaussian elimination reduces the system to row-echelon form, while Gauss-Jordan elimination reduces it to reduced row-echelon form.

In Gaussian elimination, the system of equations is transformed by using elementary row operations such as multiplying a row by a constant, adding a multiple of one row to another, or swapping two rows. This process continues until the system is in row-echelon form, where the leading coefficient of each row is to the right of the leading coefficient of the row above it.

Gauss-Jordan elimination takes this process one step further by continuing to reduce the system until it is in reduced row-echelon form, where the leading coefficient of each row is the only non-zero entry in its column. This method is useful for finding the complete solution to a system of equations, including any free variables.

In summary, vector dot product is a mathematical operation that takes two vectors and produces a scalar quantity, while Gaussian and Gauss-Jordan elimination are methods used to solve systems of linear equations by transforming them into simpler forms. I hope this helps to clarify these concepts for you.