Math Problem Statement

Ánh xạ nào là tuyến tính ?

Ánh xạ nào là tuyến tính ?

Solution

Let’s analyze the mappings to determine which are linear transformations. A mapping T:RnRmT: \mathbb{R}^n \to \mathbb{R}^m is linear if it satisfies two conditions:

  1. Additivity: T(u+v)=T(u)+T(v)T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) for all vectors u,vRn\mathbf{u}, \mathbf{v} \in \mathbb{R}^n.
  2. Homogeneity: T(cu)=cT(u)T(c \mathbf{u}) = cT(\mathbf{u}) for all scalars cc and vectors uRn\mathbf{u} \in \mathbb{R}^n.

Let’s examine each transformation one by one:


1. T1(x1,x2)=(x1+x2,x12x2)T_1(x_1, x_2) = (x_1 + x_2, x_1 - 2x_2)

This transformation can be written as:

\begin{bmatrix} 1 & 1 \\ 1 & -2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}.$$ Since $$T_1$$ is a matrix transformation, it is **linear**. --- ### **2. $$T_2(x_1, x_2) = (x_1 - x_2, 1)$$** The output contains a constant term $$1$$ in the second component. This constant term prevents $$T_2$$ from satisfying the additivity and homogeneity conditions. Therefore, $$T_2$$ is **not linear**. --- ### **3. $$T_3(x_1, x_2, x_3) = (x_1 + x_2 - x_3, 0)$$** This transformation can be written as: $$T_3(x_1, x_2, x_3) = \begin{bmatrix} 1 & 1 & -1 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}.$$ Since $$T_3$$ is a matrix transformation, it is **linear**. --- ### **4. $$T_4(x_1, x_2) = 2x_1$$** This transformation maps $$\mathbb{R}^2 \to \mathbb{R}$$ and can be written as: $$T_4(x_1, x_2) = \begin{bmatrix} 2 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}.$$ Since $$T_4$$ is a matrix transformation, it is **linear**. --- ### **Conclusion** The linear transformations are: 1. $$T_1$$, 2. $$T_3$$, 3. $$T_4$$. The transformation $$T_2$$ is not linear. --- Would you like a deeper explanation or examples of linearity? --- ### 5 Related Questions 1. What are the implications of a mapping being linear in terms of geometric transformations? 2. How can you test linearity for mappings that are not matrix transformations? 3. Why does adding a constant term violate linearity? 4. What are some real-world applications of linear transformations? 5. Can a mapping still be useful if it’s not linear? Why? ### Tip A quick way to check linearity is to express the mapping as a matrix multiplication. If it fits this form, it’s linear!

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Math Problem Analysis

Mathematical Concepts

Linear Transformations
Matrix Representation

Formulas

T1(x1, x2) = (x1 + x2, x1 - 2x2)
T2(x1, x2) = (x1 - x2, 1)
T3(x1, x2, x3) = (x1 + x2 - x3, 0)
T4(x1, x2) = 2x1

Theorems

Linearity Conditions: Additivity and Homogeneity

Suitable Grade Level

Undergraduate Mathematics