Question

Prove Theorem 3 as follows: Given an mxn matrix A, an element in Col A has the form Ax for some x in R". Let Ax! Aw represent any two vectors in Col A.

a. Explain why the zero vector is in Col A.
b. Show that the vector Ax + Aw is in Col A.
c. Given a scalar c, show that c(Ax) is in Col A

Answer 1

To prove Theorem 3, we establish that the zero vector is in Col A, the sum of any two vectors in Col A is also in Col A, and multiplying any vector in Col A by a scalar results in another vector in Col A.

Step-by-step explanation:

To prove Theorem 3 for an mxn matrix A, consider the following points:

1. The zero vector is in Col A because when the matrix A multiplies the zero vector in Rn, the resulting vector is the zero vector in Rm.
2. For vectors Ax and Aw in Col A, their sum Ax + Aw is also in Col A because matrix multiplication is distributive over vector addition, which implies A(x+w) = Ax + Aw.
3. Given a scalar c, the vector c(Ax) is in Col A because matrix multiplication is compatible with scalar multiplication, so A(cx) = c(Ax).

These properties demonstrate that Col A is closed under addition and scalar multiplication, confirming that Col A is a subspace of Rm.