Dry goods|fine notes of MIT linear algebra course [Lesson 1]

1

Knowledge summary

At the beginning of this section, we will learn about linear algebra together. In the first section, we will start with solving equations. One of the applications of learning linear algebra is to solve complex equation problems. One of the cores of this section is the knowledge of row and column images. Angle solution equation.

2

The basis of geometric interpretation of equations

2.1 Two-dimensional line image

We first use an example to solve the equation from the line image perspective:

We first write the equation in matrix form in rows:

Coefficient matrix (A): Extract the equation coefficients by rows to form a matrix.
Unknown vector (x): Extract the unknowns of the equation and form a vector in columns.
Vector (b): Extract the results on the right side of the equal sign in columns to form a vector.

Next, we use row images to solve this equation: the
so-called row image is to take one row at a time to form the equation on the coefficient matrix, and draw on the coordinate system. It is no different from the process of drawing and solving equations we learned in elementary mathematics.

2.2 Two-dimensional column image

Next, we use column images to solve this equation:

Find the appropriate x, y such that x times (2,-1) + y times (-1,2) to get the final vector (0,3). Obviously, it can be seen that 1 times (2,-1) + 2 times (-1,2) meets the condition.

Reflected on the image, the result is obviously correct.

3

Extension of geometric interpretation of equations

3.1 high-dimensional line image


if the image to draw the line, it is clear that this is one of the three planes intersect get a little, we would like to see the direct nature of this point can be described as difficult.

A more reliable way of thinking is to first connect two of the planes so that they intersect a straight line. After studying the point at which the straight line intersects the plane, the point coordinates are the solution of the equation.

This solution process may be reasonable for 3D, but what about 4D? What about five dimensions or even higher dimensions? It is intuitively difficult to directly draw higher-dimensional images, and such line images are subject to more and more restrictions.

3.2 High-dimensional column image

The left side is the linear combination, and the right side is the result of the appropriate linear combination. In this way, the idea is much clearer. "Finding a linear combination" becomes the key to understanding the problem.

Obviously this question is a special case, we only need to take x = 0, y = 0, and z = 1. The result is obtained, which is not obvious in the line image.

Of course, the reason why we recommend using column images to solve equations is because this is a more systematic solution method, that is, looking for linear combinations, instead of looking for points that are difficult to see after drawing each row of equations.

Another advantage is that if we change the final result b, for example, in this question,

Then it is enough for us to find a linear combination for 2 −1 1 0 −3 4 −3, but what if we use line images? That means we have to completely redraw the three plane images. In terms of simplicity, the two methods are superior to each other.

In addition, it should be noted that can the matrix equation Ax = b be solved for any b? That is, for the 3*3 coefficient matrix A, can the linear combination of its columns cover the entire three-dimensional space?

For the example we gave, it must be possible, and the 2*2 example above can also cover the entire plane, but some matrices are not acceptable.

For example, the three column vectors themselves constitute a plane, so the vector combined by such three vectors can only move on this plane, and certainly cannot cover a 2 −1 1 three-dimensional space.

These three vectors constitute a plane. .

3.3 Matrix multiplication

4

Learning perception

This part of the content is the initial reference to the concept of linear algebra. Starting from solving equations, introducing the concept of column space, you can find that from the perspective of column space, it is more scientific to change the equation to be solved into a linear combination of column vectors. Introduced matrix multiplication, this part focuses on understanding.

I hope it helps everyone~

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