Linear Mapping and Rank-Nullity Verification in R^3

Math Problem Statement

Determine the linear mapping T: R^3 -> R^3 which maps the basis vectors (0,1,1), (1,0,1), (1,1,0) of R^3 to (1,1,2), (1,1,1), (1,1,1) respectively. Verify that dim ker T + dim Im T = 3.

Solution

This is a linear algebra question about determining a linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^3$ , mapping the given basis vectors of $\mathbb{R}^3$ , and verifying properties of $T$ . Let's break it down step by step.

Problem Summary:

Determine the linear mapping $T$ :
- Basis vectors $(0,1,1), (1,0,1), (1,1,0)$ are mapped to $(1,1,2), (1,1,1), (1,1,1)$ , respectively.
- Express $T$ as a matrix transformation.
Verify $\dim(\ker(T)) + \dim(\text{Im}(T)) = 3$ :
- This is the rank-nullity theorem: $\dim(\ker(T)) + \dim(\text{Im}(T)) = \dim(\mathbb{R}^3)$ .

Step-by-Step Solution:

1. Representing $T$ as a matrix:

Let $T$ be represented by a $3 \times 3$ matrix $A$ . The action of $T$ on any vector is $T(\mathbf{v}) = A\mathbf{v}$ .

We use the given basis and their images to determine $A$ :

$T(0,1,1) = (1,1,2)$
$T(1,0,1) = (1,1,1)$
$T(1,1,0) = (1,1,1)$

Write these as equations:

A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad A \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}.$$ Using these, we can determine the columns of $$A$$ by solving for the coefficients. #### 2. Constructing $$A$$: Each column of $$A$$ corresponds to the image of a standard basis vector: $$A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 2 & 1 & 1 \end{bmatrix}.$$ #### 3. Verifying $$\dim(\ker(T)) + \dim(\text{Im}(T)) = 3$$: - **Image (Rank):** The rank of $$A$$ is the number of linearly independent columns. Here, the first and second columns are identical, so the rank is 2. $$\dim(\text{Im}(T)) = 2.$$ - **Kernel (Nullity):** Using the rank-nullity theorem: $$\dim(\ker(T)) = 3 - \dim(\text{Im}(T)) = 1.$$ Thus, $$\dim(\ker(T)) + \dim(\text{Im}(T)) = 1 + 2 = 3$$, as required. --- ### Final Answer: 1. The matrix representation of $$T$$ is: $$A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 2 & 1 & 1 \end{bmatrix}.$$ 2. The verification shows that $$\dim(\ker(T)) + \dim(\text{Im}(T)) = 3$$, satisfying the rank-nullity theorem. --- Would you like a detailed explanation of any part of this process? Let me know! --- ### 5 Related Questions: 1. How do you compute the rank of a matrix? 2. What is the geometric interpretation of the kernel and image of a linear transformation? 3. How do you find the kernel of a matrix? 4. What happens when a basis vector is mapped to zero in a transformation? 5. How can you verify that a given transformation is linear? ### Tip: When finding the matrix of a linear transformation, always write the images of the basis vectors in terms of the columns of the matrix!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Linear Transformations
Matrix Representation

Formulas

T(v) = Av, where A is a 3x3 matrix

Theorems

Rank-Nullity Theorem

Suitable Grade Level

Undergraduate

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