The red buttons will take you to selected solutions to the HW problems, posted after the homework has been graded. Generally speaking, these will be the ones that students had the most difficulty with. The solutions are in PDF format.

**A. ** Finish the proof given in class that the inverse of a linear function is also a linear function, by proving:

Let T:**R**^{n} → **R**^{n}
be a linear function which is invertible, and let S be its inverse. Then
for all **x** in **R**^{n} , and all real numbers r, we have S(r**x**) = rS(**x**).

**B. **
Let T:**R**^{n} → **R**^{n}
be a linear function that is represented by a matrix A. Suppose that the matrix A is invertible, ie that there is a matrix B with AB = BA = I_{n}. Prove that T is one-to-one and onto. (This is the converse of what we proved in class.)

**C. ** Prove that if a vector **v** in **R**^{n} satisfies **v · w** = 0 for all **w** in **R**^{n}, then **v**= **0**. Combine this with the result of problem 1 (p. 209) to prove that for any matrix A, and any vectors **v** and **w**, we have (image(A))^{¬} = ker(A^{T}). (The ¬ sign is supposed to be the upside down T denoting orthogonal complement, but I couldn't get it to some out on the web page.)

Back to the class homepage.