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Matrix Arithmetic

Part 2: The product Ax as a linear combination

In this section we need some special vectors called standard unit vectors. We define the four-dimensional standard unit vectors in the the worksheet -- the symbols e1, e2, e3, and e4 are commonly used for these vectors. We continue to use the matrix A defined in Part 1.

  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.

  3. Express Ax as a linear combination of columns of A. [Hint: Multiply your expression from Step 2 by A.] Check by computing both Ax and your linear combination of columns.

  4. 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.

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