Take 2-dimensional space as an example, there is a set of basis vectors. Any vector in this two-dimensional space can be expressed by this set of basis vectors, so this two-dimensional space is a space expanded by this set of basis vectors. The specific performance is as follows:

Where is any real number and is also a value.

Any vector in space can be obtained only by scaling and adding the base vector, which also shows the special importance of vector addition and number multiplication.

therefore

Naturally, there are countless groups of such basis vectors. In two-dimensional space, we usually choose the above as the base vector.

Transformation is actually equivalent to a function. In this case, the function inputs a vector and outputs a vector.

The vector dimensions of input and output can be different.

The reason why it is defined by transformation rather than function is because transformation emphasizes a moving process. For example, in two-dimensional space, we can imagine that the vector moves to other positions in the space after linear transformation.

There are two kinds of transformations: linear transformation and nonlinear transformation. This section talks about linear transformation and its relationship with matrix.

Imagine a vector as an arrow, then linear transformation refers to the movement of the vector with the starting point at the origin in different spaces, keeping the multiplication and addition invariance of the vector number.

This different space can be understood as

For example, a three-dimensional vector is converted into a three-dimensional vector by linear transformation.

Or a three-dimensional vector is converted into a two-dimensional vector by linear transformation.

1 mentioned above is actually a special case of 2. If the transformed space dimensions are different, the basis vector of the space definition must have changed.

Intuitively, we can use

Two conditions for linear transformation.

As we know, linear transformation is to move all vectors in space to a new position. In this process, the starting point of the vector remains unchanged. So how do you track any converted vector?

As we know from the last section, a vector is actually a linear combination of base vectors, and any vector can be represented by base vectors.

How do you know the transformation of the basis vector? In two-dimensional space, we only need to observe this set of basis vectors. The coefficient of the basis vector after linear transformation is the coefficient of the basis vector before linear transformation, that is, the coordinates before linear transformation.

known

That is, after linear transformation, it becomes, that is, at this time, it is correspondingly transformed into,, and

certificate

According to the definition of linear transformation above:

therefore ...

So as long as we know the coordinates of the transformed base vector, we can perform linear transformation.

Now assume that the basis vector after linear transformation is known.

Borrow the known conditions in the above proof.

,

Then we "package" the coordinates into a new grid, which we call a matrix.

Seeing this, everyone should understand that the original matrix is the splicing of basis vectors after linear transformation.

In daily application, a matrix is usually given, so this section begins with the assumption that the transformed basis vector is known to be true and is an element of the matrix.

Then any transformation vector in the space can be represented by the basis vector.

Please look at the following example:

There is a matrix and a vector. Under the action of the matrix, the new vector coordinates (moved to a new position) are as follows:

Please read and follow the article carefully.

This form is similar, equivalent to the coefficient of the base vector,

It is the basis vector after linear transformation.

Therefore, the intuitive explanation of matrix and vector multiplication is as follows:

Because matrix represents the linear transformation of space, matrix multiplication means that the transformed base vector is linearly transformed again, that is, the original space is linearly transformed twice.

The effect of two transformations is equivalent to the effect of one transformation of 1 matrix obtained by multiplying two matrices.

The main content comes from the essence of linear algebra of Upmaster @3Blue 1Brown.