矩阵加法原理及水平切变变换中向量运算的技术问询

Hey there! Let's break this down clearly for you, since you're working through a bigger math problem and already know the answer but want to solidify the basics.

Great news—your运算 logic is 100% correct! Let's recap to confirm:

• You were told to keep the standard basis vector $e_1 = \begin{bmatrix}1 \ 0 \ \end{bmatrix}$ unchanged
• For $e_2$, you needed to compute $e_2 + 3e_1$, which you did step-by-step perfectly:
  $e_2 + 3e_1 = \begin{bmatrix}0 \ 1 \ \end{bmatrix} + 3*\begin{bmatrix}1 \ 0 \ \end{bmatrix} = \begin{bmatrix}0 \ 1 \ \end{bmatrix} + \begin{bmatrix}3 \ 0 \ \end{bmatrix} = \begin{bmatrix}3 \ 1 \ \end{bmatrix}$

The resulting shear transformation matrix is just these new basis vectors arranged as columns: $\begin{bmatrix}1 & 3\ 0 & 1\ \end{bmatrix}$. This matrix will apply a horizontal shear to any vector in the plane—meaning every point's y-coordinate stays the same, while its x-coordinate gets shifted by 3 times its y-value.

Matrix addition is one of the foundational operations in linear algebra, and it has two non-negotiable rules to remember:

• Rule 1: Same dimensions only You can only add two matrices if they have exactly the same number of rows and columns. A 2x2 matrix can't add to a 2x1 vector, for example—they don't match up.
• Rule 2: Add corresponding elements For each position (row, column) in the matrices, you add the element from the first matrix to the element in the exact same position in the second matrix.

Let's use a concrete 2x2 matrix example to make this tangible:
If $A = \begin{bmatrix}a_{11} & a_{12}\ a_{21} & a_{22}\ \end{bmatrix}$ and $B = \begin{bmatrix}b_{11} & b_{12}\ b_{21} & b_{22}\ \end{bmatrix}$, then:
$A + B = \begin{bmatrix}a_{11}+b_{11} & a_{12}+b_{12}\ a_{21}+b_{21} & a_{22}+b_{22}\ \end{bmatrix}$

When you added $e_2 + 3e_1$, you were actually doing vector addition—and vectors are just matrices with a single column (or row). The scalar multiplication step ($3*e_1$) is a prerequisite here: you first scale every element of $e_1$ by 3, then add the corresponding elements to $e_2$. That's exactly how linear combinations (like this shear transformation) work!

Let's visualize this shear in the 2D coordinate plane to make it click:

1. Original coordinate system
   • $e_1$ sits at (1, 0), pointing along the x-axis
   • $e_2$ sits at (0, 1), pointing straight up the y-axis
   • Together, these form the standard right-angle grid we all learn in geometry, represented by the identity matrix $\begin{bmatrix}1 & 0\ 0 & 1\ \end{bmatrix}$
2. After shear transformation
   • $e_1$ stays exactly where it is—no change
   • The new $e_2$ moves to (3, 1): think of this as taking the original (0,1) vector and sliding it 3 units to the right (since we added 3 times the x-axis vector $e_1$)
   • The entire grid tilts! The y-axis, which was vertical, now becomes a line from (0,0) to (3,1) (slope 1/3). Every point in the plane shifts accordingly: for any point (x, y), its new position is (x + 3y, y). Imagine grabbing the top of a grid paper and pulling it to the right—you'll get that slanted parallelogram grid, which is exactly this horizontal shear effect.

内容的提问来源于stack exchange，提问作者Liath