Row of a matrix is a vector along the same degree of freedom

In summary, the conversation discusses different ways of interpreting matrices, such as viewing them as arrays of vectors and as operators that convert one vector into another vector. The role of transpose vectors and the geometric representation of matrices are also mentioned.
  • #1
Gear300
1,213
9
A particular introduction to matrices involved viewing them as an array/list of vectors (column vectors) in Rn. The problem I see in this is that it is sort of like saying that a row of a matrix is a vector along the same degree of freedom (elements of the same row are elements of different vectors all in the same dimension). So from this, technically, the scalar product of a column vector v and row1 of a matrix A should only exist as a product between the elements of row1 of the matrix A and row1 of the column vector v...which doesn't seem right (since matrix-vector multiplication Av is defined as a column vector of dot products between the vector v and rows of A). How would one geometrically interpret a matrix?
 
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  • #2
Gear300 said:
… So from this, technically, the scalar product of a column vector v and row1 of a matrix A should only exist as a product between the elements of row1 of the matrix A and row1 of the column vector v...

which doesn't seem right (since matrix-vector multiplication Av is defined as a column vector of dot products between the vector v and rows of A).

Hi Gear300! :smile:

Technically, row vectors are transpose vectors,

so the first row of A is is not the vector a1 (say), but the transpose vector a1T.

Then your scalar product in matrix form is (a1T)v,

but in ordinary form is a.v :wink:
How would one geometrically interpret a matrix?

Dunno :redface:, except that I always think of a matrix as being a rule that converts one vector into another vector. :smile:
 
  • #3


I understand matrices as operators, like you may have basis vectors (x,y,z all normal), you can have basis functions (such as sin(nx) for n=1,2,3...) which any periodic function can be decomposed into using Fourier series. Then the function can be represented completely by a column vector containing the amplitude of each frequency, and the matrix multiplication will output a new function. So in that sense you might represent a matrix geometrically by a series of "before and after" functions!

Then again this is just fun speculation... the most fundamental way of expressing a matrix geometrically is probably by a discrete 2-D plot, f(m,n) = the (m,n)th element of the matrix.
 
  • #4


I see...good stuff so far...Thanks for all the replies.
 
  • #5
Hi Gear300! :smile:

I've found http://farside.ph.utexas.edu/teaching/336k/lectures/node78.html#mom" on john baez's excellent website which shows the relation between the ordinary and the matrix representation of the same equation, and where the transpose fits in …

if ω is the (instantaneous) angular velocity, then the rotational kinetic energy is both:

1/2 ω.L
and (without the "dot")
1/2 ωTL = 1/2 ωTÎω

where L is the (instantaneous) angular momentum, and Î is the moment of inertia tensor, both measured relative to the centre of mass.
 
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  • #6


Thanks a lot for the link tiny-tim.

So, its sort of like saying that for an m x p matrix with n being a dimensional space, you could label the rows (going down) from n = 1 to n = m and you could also label the columns (going across) from n = 1 to n = p, right?
 
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1. What does it mean for a row of a matrix to be a vector along the same degree of freedom?

When we talk about the "degree of freedom" of a vector, we are referring to the number of independent components or variables that it has. In a matrix, each row represents a vector, and the "degree of freedom" refers to the number of elements in that vector. Therefore, a row of a matrix is a vector along the same degree of freedom if it has the same number of elements as the other rows in the matrix.

2. Why is it important to understand the concept of a row of a matrix being a vector along the same degree of freedom?

Understanding this concept is important because it allows us to perform operations on matrices, such as addition and multiplication, while maintaining the same number of variables. This is crucial in various fields, including physics, engineering, and computer science, where matrices are used to represent complex systems and equations.

3. Can a row of a matrix have a different degree of freedom than the other rows?

No, in a matrix, all rows must have the same number of elements. Otherwise, the matrix would not be well-defined, and we would not be able to perform operations on it. This is because the operations are carried out element-wise, meaning each element in a row must correspond to the same element in the other rows.

4. How does the concept of a row of a matrix being a vector along the same degree of freedom relate to linear independence?

Linear independence refers to a set of vectors that cannot be written as a linear combination of each other. The number of vectors in a set that are linearly independent is equal to the "degree of freedom" of the set. Therefore, in a matrix, the number of linearly independent rows is equal to the number of elements in each row, and they are all vectors along the same degree of freedom.

5. Are there any real-world applications of the concept of a row of a matrix being a vector along the same degree of freedom?

Yes, there are many real-world applications of this concept. For example, in physics, matrices are used to represent physical systems, and understanding the degree of freedom of the rows can help us solve complex equations and understand the behavior of these systems. In computer science, matrices are used in machine learning algorithms, and the degree of freedom of the rows can affect the accuracy and performance of these algorithms. Additionally, in engineering, matrices are used to design and analyze structures, and the degree of freedom of the rows is crucial in determining the stability and strength of these structures.

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