According to the introduction in the linear algebra course and textbook, left multiplication of a matrix is a row transformation, and right multiplication of a matrix is a column transformation.
But for someone like me who lacks abstract mathematical ability, such explanations are always not specific enough, and it seems to be a stumbling block to understanding.
So I spent several hours pondering over this simple concept, and finally got some clues.
Let's talk about what transformation is first.#
As mentioned earlier, left multiplication of a matrix is a row transformation, and right multiplication of a matrix is a column transformation. So who is being transformed by this transformation? And how does this operation form a transformation?
We know that the common form of matrix-vector multiplication is as follows:
Ax=b
where A is the matrix, x is the column vector, and b is also a column vector.
And the above form is similar to a concept we have learned in the past, which is a function. Recall the form of a function:
f(x)=b
Based on the knowledge we learned in high school, x and b should be some numerical values. But in advanced mathematics, the scope of discussion has been greatly expanded to vectors, which means that the input of a function does not necessarily have to be a numerical value, and the output of a function does not necessarily have to be a numerical value either.
The input and output of a function can be a vector. This is actually an impression of a multivariable function, and Ax=b is just one expression of a multivariable function.
Actually, the above statement has some conceptual inaccuracies in terms of details. That is, f(x)=b, or Ax=b should be an equation instead of a function. f(x)=Ax is the expression of a function, and b is a value in the range. Expanding on this topic is another small topic, which can be discussed slowly in the future if there is time.
Continuing with the original topic, we have learned that A is actually a transformation, which transforms the vector x into the vector b. But this understanding is still too abstract. How does the multiplication of vector x and matrix A become vector b?
Here we need to review the concept of a function. According to the standard concept of a function, a function is a mapping from numbers in one set to numbers in another set. By analogy, for a multivariable function, a function is a mapping from vectors in one space to vectors in another space (here, another question arises, what is space, which is a big topic and can be temporarily understood as a set composed of all vectors).
In other words, the x vector in one space is transformed through A and mapped to the b vector in another different space. This is a transformation.