1. Prove that the space is a subspace or demonstrate which of the requirements for a subspace are violated:

a. The set of vectors in R2 whose components are positive or zero.

b. The set of functions continuous on the open interial from zero to one.

2. Given the system:

x + 2y - 2z = b₁

x + 5y - 4z = b₂

x + 9y - 8z = b₃

a. Find the solvability condition.

b. Find the basis for the column space.

c. Find a particular solution for when b =

d. Find the basis for the null space.

e. Write the complete solution to the system.

f. Find the basis for the row space.

g. Find the basis for the left null space.

3. Identify a vector that lies in each of the fundamental spaces identified in the previous problem. Use these to illustrate the relationships between pairs of the fundamental spaces.

4. Given

determine if there is a left inverse or a right inverse (support required) and calculate it.

5. Prove that for a linear transformation T, T(0) = 0.

6. Write a matrix that calculates the slope of a linear equation (L(x) = l₁ + l₂x). Show that the resulting transformation is linear.