Matrix Operations

Part 2: The product Ax as a linear combination

The symbols e₁, e₂, e₃, and e₄ are commonly used for the standard unit vectors in four dimensions. These vectors are defined in the worksheet.

1. Compute each of the products Ae₁, Ae₂, Ae₃, and Ae₄. Compare the products to the original matrix A, and then generalize what you see by answering this question: If A were any m x n matrix and ej were an n-dimensional standard unit vector, how could you characterize the product Aej?
2. For the vector x defined in the worksheet, express x as a linear combination of standard unit vectors. Express Ax as a linear combination of columns of A. Check by computing both Ax and your linear combination of columns.
3. What should the vector x be if you want the product Ax to be the difference between the second and third columns of A? Test your answer in the worksheet.