Why Is Matrix Multiplication Defined the Way It Is?

In summary, The conversation revolves around the usefulness of matrix multiplication and the reason behind its commonly used definition. It is mentioned that understanding this concept is essential, as it can represent any linear transformation and is necessary for the calculation of matrix products. The commonly used definition is chosen to ensure that the product of two linear transformations is equivalent to the product of their respective matrix representations.
  • #1
jacobrhcp
169
0
Something that has bothered me in my linear algebra class was that I learned a lot of techniques but didn't learn why they worked, or what they were useful for.

One of the things is this: why is matrix multiplication so useful in the way it's defined, and not in any other way? Of all the ways we could define the components of a matrix after multiplication, why does the commonly used way turn out to be so great?

I feel this is such a basic property that I should've learned it a long time by now, but I haven't, and I feel sorry about that.
 
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  • #2
every linear transformation can be represented by a matrix

so take f, g linear and let A, B be their matrix representations respectively, then the matrix representation of fog is the matrix product AB, so it's defined the way it is(however strange it seems at first) so that this works
 

What is matrix multiplication?

Matrix multiplication is a mathematical operation where two matrices are multiplied together to create a new matrix. It involves multiplying each element of one matrix by the corresponding element in the other matrix and then summing up the products.

What are the requirements for matrix multiplication?

In order for two matrices to be multiplied together, the number of columns in the first matrix must match the number of rows in the second matrix. This means that an m x n matrix can only be multiplied by an n x p matrix, resulting in an m x p matrix.

What is the difference between scalar multiplication and matrix multiplication?

Scalar multiplication involves multiplying each element of a matrix by a single number, while matrix multiplication involves multiplying two matrices together. Scalar multiplication results in a matrix with the same dimensions as the original matrix, while matrix multiplication results in a new matrix with different dimensions.

Why is matrix multiplication important?

Matrix multiplication is important in many fields of science and engineering, such as physics, computer science, and economics. It allows us to solve systems of linear equations, transform data, and perform calculations in higher dimensions.

What are some properties of matrix multiplication?

Matrix multiplication is associative, meaning that the order in which matrices are multiplied does not change the end result. It is also not commutative, meaning that the order of the matrices does matter. Additionally, the identity matrix serves as the neutral element for matrix multiplication, and multiplying a matrix by the identity matrix results in the original matrix.

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