Math Problem Statement

solve this but write question first

Let T: R^3 -> R^3 be the linear transformation satisfying: T([1, 0, 0]) = [1, 1, 0], T([0, 1, 0]) = [1, 0, 1], and T([0, 0, 1]) = [-1, 1, 1]. Find T([x, y, z]). Also, find the rank of the 3x3 matrix A such that T(X) = AX.

Solution

The question from the image is as follows:

Let T:R3→R3T: \mathbb{R}^3 \to \mathbb{R}^3 be the linear transformation satisfying:

T\left(\begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix}\right) = \begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}, \quad T\left(\begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}\right) = \begin{bmatrix}-1 \\ 1 \\ 1\end{bmatrix}.$$ Find $$T\left(\begin{bmatrix}x \\ y \\ z\end{bmatrix}\right)$$. Also, find the rank of the $$3 \times 3$$ matrix $$A$$ such that $$T(X) = AX$$. --- Now let's solve it. --- ### **Step 1: Write $$A$$, the transformation matrix** The transformation matrix $$A$$ is constructed by placing the images of the standard basis vectors as columns: $$A = \begin{bmatrix} T\left(\begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}\right) & T\left(\begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix}\right) & T\left(\begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}\right) \end{bmatrix}.$$ Substituting the given values: $$A = \begin{bmatrix} 1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix}.$$ --- ### **Step 2: Find $$T\left(\begin{bmatrix}x \\ y \\ z\end{bmatrix}\right)$$** To compute $$T\left(\begin{bmatrix}x \\ y \\ z\end{bmatrix}\right)$$, multiply $$A$$ with the column vector $$\begin{bmatrix}x \\ y \\ z\end{bmatrix}$$: $$T\left(\begin{bmatrix}x \\ y \\ z\end{bmatrix}\right) = A \begin{bmatrix}x \\ y \\ z\end{bmatrix}.$$ Performing the matrix multiplication: $$T\left(\begin{bmatrix}x \\ y \\ z\end{bmatrix}\right) = \begin{bmatrix} 1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix} \begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix} x + y - z \\ x + z \\ y + z \end{bmatrix}.$$ So: $$T\left(\begin{bmatrix}x \\ y \\ z\end{bmatrix}\right) = \begin{bmatrix}x + y - z \\ x + z \\ y + z\end{bmatrix}.$$ --- ### **Step 3: Find the rank of $$A$$** The rank of a matrix is the number of linearly independent rows (or columns). To find the rank of $$A$$: $$A = \begin{bmatrix} 1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix}.$$ Perform row reduction: 1. Subtract the first row from the second row: $$R_2 = R_2 - R_1 \implies \begin{bmatrix} 1 & 1 & -1 \\ 0 & -1 & 2 \\ 0 & 1 & 1 \end{bmatrix}.$$ 2. Add $$R_2$$ to $$R_3$$: $$R_3 = R_3 + R_2 \implies \begin{bmatrix} 1 & 1 & -1 \\ 0 & -1 & 2 \\ 0 & 0 & 3 \end{bmatrix}.$$ The matrix is now in row-echelon form, and there are 3 non-zero rows. Thus, the rank of $$A$$ is: $$\text{Rank}(A) = 3.$$ --- ### **Final Answers** 1. $$T\left(\begin{bmatrix}x \\ y \\ z\end{bmatrix}\right) = \begin{bmatrix}x + y - z \\ x + z \\ y + z\end{bmatrix}$$. 2. The rank of $$A$$ is $$3$$. --- Would you like further explanation or assistance with similar problems? Here are five related questions: 1. How can we verify that the columns of $$A$$ are linearly independent? 2. What does the rank of $$A$$ tell us about the mapping $$T$$? 3. How do we find the nullity of $$T$$ using the rank-nullity theorem? 4. What would change if $$T$$ mapped $$\mathbb{R}^3$$ to $$\mathbb{R}^2$$ instead? 5. How can we determine if $$T$$ is an isomorphism based on $$A$$? **Tip:** Always verify your row-reduction steps carefully to avoid errors when computing rank!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Linear Transformations
Matrix Multiplication
Rank of a Matrix

Formulas

Matrix representation of a linear transformation: A = [T(e1), T(e2), T(e3)]
Matrix-vector multiplication: T(X) = AX
Rank of a matrix: Number of linearly independent rows or columns

Theorems

Rank-Nullity Theorem

Suitable Grade Level

Undergraduate Level (First-Year or Second-Year)

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