What does a row matrix represent geometrically?

[tellmesomething]

Homework Statement

What does a row matrix represent geometrically ?

Relevant Equations

None

I understand that a column matrix of order n×1 is taken as a vector with each element representing the x coordinate (x ##\hat i ##) y coordinate(y ##\hat j ##) z coordinate(z ##\hat k ##) ... n coordinate (n × unit vector in nth dimension) of that vector.

I also understand that a matrix of order m×n , each column represents the linearly transformed vector of unit vectors leading up to the nth dimension.

What I dont get is what does a row matrix represent?
Does it represent the linearly transformed unit vectors but in this case it has all been squished into one dimension?

 

[nuuskur]

As you say, an ##m\times n## matrix over ##\mathbb R##, say, represents a linear map ##\mathbb R^n \to\mathbb R^m##. Hence, a row would correspond to a linear map ##\mathbb R^n\to\mathbb R##. A projection, for instance.

 

[FactChecker]

You should first say that, you are using the matrix, ##M##, as a multiplier on the left of a vector, ##\vec{v}##: ##M \vec{v}##, rather than on the right.

What I dont get is what does a row matrix represent?
Does it represent the linearly transformed unit vectors but in this case it has all been squished into one dimension?

Yes, you are correct. In the case of a general ##m \times n## matrix, each of the ##m## rows represents a linear combination of the elements of any ##n \times 1## vector that it multiplies on the left. That is true no matter what the row dimension, ##m##, of the matrix is.

 

[WWGD]

A matrix in its most general sense, is a rectangular array of Mathematical objects that takes meaning depending on context. It can be an adjacency matrix for a graph, a correlation matrix between two Random Variables, a linear transformation.

 

[Mark44]

What I dont get is what does a row matrix represent?

Also a vector. You can think of a dot product for vectors in ##\mathbb R^3## as a product of a row matrix (on the left) and a column vector (on the right).

For example,
##\begin{bmatrix} 1 & 3 & 2 \end{bmatrix}\begin{bmatrix} 2 \\ 3 \\ 6 \end{bmatrix} = 1\cdot 2 + 3\cdot 3 + 2\cdot 6 = 2 + 9 + 12 = 23##

Be aware that if you have a column vector on the left and a row vector on the right you get something different. For the example above if the vectors are switched, you get a 3 x 3 matrix.

 

[DaveE]

https://ocw.mit.edu/courses/18-06-l...es/lecture-10-the-four-fundamental-subspaces/

 

[tellmesomething]

You should first say that, you are using the matrix, ##M##, as a multiplier on the left of a vector, ##\vec{v}##: ##M \vec{v}##, rather than on the right.

Yes, you are correct. In the case of a general ##m \times n## matrix, each of the ##m## rows represents a linear combination of the elements of any ##n \times 1## vector that it multiplies on the left. That is true no matter what the row dimension, ##m##, of the matrix is.

I see thankyou for your reply. :) additionally do you have any recommended books to further understand the different types of matrices its applications and its vector interpretation?

 

[FactChecker]

do you have any recommended books to further understand the different types of matrices its applications and its vector interpretation?

None that stand out from the others. You should know that there are a great variety of uses and interpretations of different types of matrices, some with very good geometric interpretations, and others more obscure. Linear functions can be represented by matrices, but there can be disadvantages of using a matrix which makes a representation depend on a particular basis and coordinate system.

 

[WWGD]

I see thankyou for your reply. :) additionally do you have any recommended books to further understand the different types of matrices its applications and its vector interpretation?

I liked Steven Leon's Linear Algebra book. You may complement it with plenty of YouTube videos. Several from Prof Steven Brunton from University of Washington alone, include both theory and applications of SVD to face recognition and general Machine Learning

https://www.google.com/search?q=You...yAEAmAIAoAIAmAMAkgcAoAcA&sclient=gws-wiz-serp