To determine the nullity and rank of the given linear transformation:

1. Write the transformation in matrix form: The transformation maps $\mathbf{x} = [x_1, x_2, x_3, x_4]^T$ to $\mathbf{y} = [y_1, y_2, y_3, y_4]^T$, where:

   \begin{bmatrix} x_1 + x_2 \\ x_2 + x_3 \\ x_3 + x_4 \\ x_4 + x_1 \end{bmatrix}$$ This can be represented in matrix form as: $$\mathbf{y} = A \mathbf{x},$$ where $$A$$ is the coefficient matrix: $$A = \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 \end{bmatrix}.$$

2. Find the rank of the matrix $A$: To compute the rank of $A$, reduce it to row echelon form.

   \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 \end{bmatrix}$$ Perform row reduction: - Subtract row 1 from row 4: $$R_4 \to R_4 - R_1 \implies \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & -1 & 0 & 1 \end{bmatrix}.$$ - Add row 2 to row 4: $$R_4 \to R_4 + R_2 \implies \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{bmatrix}.$$ - Subtract row 3 from row 4: $$R_4 \to R_4 - R_3 \implies \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}.$$ The matrix is now in row echelon form. The rank of $$A$$ is the number of nonzero rows, which is **3**.

3. Find the nullity: The nullity of a matrix is given by: $\text{Nullity} = n - \text{Rank},$ where $n$ is the number of columns of $A$. Here, $n = 4$, and the rank is 3. Thus: $\text{Nullity} = 4 - 3 = 1.$

Final Answer:

• Rank = $3$
• Nullity = $1$

Would you like a more detailed explanation of the row reduction or examples of how to interpret nullity and rank geometrically?

Expanding Questions:

1. Can you explain the relationship between rank and the number of independent rows or columns?
2. How does nullity relate to the solutions of the homogeneous system $A\mathbf{x} = 0$?
3. Can you describe the geometric interpretation of the rank and nullity of this transformation?
4. How would the rank and nullity change if additional rows or columns were added to the matrix?
5. What are the implications of rank and nullity in solving linear systems?

Tip:

The rank-nullity theorem states that the rank and nullity of a matrix always sum up to the number of columns. Use this as a quick check for your calculations!