- #1
Gear300
- 1,209
- 9
What would be the proof for matrix multiplication?...or just an explanation as to why its done the way its done.
Why is matrix multiplication necessary for representing linear transformations?
- Context: Undergrad
- Thread starter Gear300
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Discussion Overview
The discussion revolves around the necessity and definition of matrix multiplication in the context of representing linear transformations. It explores the foundational aspects of matrices, their multiplication, and their applications in solving systems of linear equations and understanding linear transformations.
Discussion Character
- Exploratory
- Technical explanation
- Conceptual clarification
Main Points Raised
- Some participants seek a proof or explanation for the definition of matrix multiplication.
- One participant notes that matrix multiplication is defined in a certain way but expresses uncertainty about the reasons behind that definition.
- Another participant describes matrices as rectangular arrays of data and questions how the multiplication of these arrays operates.
- It is proposed that the definition of matrix multiplication facilitates the solving of systems of linear equations and represents linear transformations, which are important in nonlinear mathematics.
- A participant explains the interpretation of a 2x2 matrix in terms of linear transformations of points in the plane, emphasizing the movement of points through successive transformations.
- Another viewpoint suggests that matrices can be seen as linear transformations from R^n to R^m, with matrix multiplication corresponding to the composition of these linear maps.
Areas of Agreement / Disagreement
Participants express varying levels of understanding and interpretation regarding matrix multiplication and its significance. There is no consensus on a definitive explanation or proof, and multiple perspectives on the topic remain present.
Contextual Notes
Some limitations include a lack of detailed proofs for the claims made, as well as differing interpretations of matrices and their operations. The discussion does not resolve the foundational questions about the definition of matrix multiplication.
Who May Find This Useful
This discussion may be useful for individuals interested in the foundational concepts of linear algebra, particularly those exploring the relationship between matrices and linear transformations.
- #2
Set Abominae
- 15
- 0
It's done like that by definition, as far as I am aware. I don't really know why it's defined like that though...
- #3
Gear300
- 1,209
- 9
I see...I haven't really studied matrices all too much, so I'm not too sure of what they are (by what I can say, a matrix is a rectangular array of data or an organization of data). I'm just wondering how multiplying two rectangular arrays of data works.
- #4
slider142
- 1,016
- 72
They're defined to multiply in that manner so that they can be used to solve systems of linear equations. The manner of multiplication is also convenient in that it easily represents a linear transformation, which is very useful in studying nonlinear mathematics. A transformation T of two objects u and v is linear if T(s*u + v) = s*T(u) + T(v). Ie., if x is a vector denoted by a column of numbers, a linear transformation of x can always be represented by matrix multiplication from the left (Ax).
- #5
Cantab Morgan
- 261
- 2
Let's think of a 2x2 matrix, which represents a linear transformation of the plane. I like to think of matrices as columns of numbers, not as rows. Then the left column of the matrix represents where the point (1,0) on the horizontal axis goes. The right column represents where the point (0,1) on the vertical axis goes.
Each time you multiply a matrix by another, you are following up one linear transformation by another. You can imagine how the (1,0) point moves after one transformation, and then how the resulting vector moves after the next. The rules for calculating products of matrices might be easier to think about that way.
Each time you multiply a matrix by another, you are following up one linear transformation by another. You can imagine how the (1,0) point moves after one transformation, and then how the resulting vector moves after the next. The rules for calculating products of matrices might be easier to think about that way.
- #6
yyat
- 315
- 0
More abstractly, if you think of matrices as linear transformations R^n->R^m, then matrix multiplication corresponds to composition of linear maps. So A(B(v))=(AB)(v), where AB is the matrix product. This requires and (easy) proof.
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