On 10/15/99, claybomi asks about Q1 of MA322-005,08 Section 3.1 and 3.2 (Oct 14): The hint on this one really confused me! Response: We talked about this in class. The 'addition' that is defined in this problem is not commutative. as the example shows. (2) fails too: take u = v = w = [1,1] and compute that u + (v+w) = [3,1] but (u +v) +w = [3,-1]. (3) holds: [0,0] = O. (4) holds: Let u = [a,b]. Then if we take -u = [-a,b], we see that u + (-u) = [a,b]+[-a,b] =[0,0]. Note however that (-u) + u = [0,-2b] which is not [0,0] unless b = 0. The remaining properties can be seen to hold in general too.

On 10/20/99, colleter asks about Q1 of MA322-005,08 Section 3.1 and 3.2 (Oct 14): I interpreted this question, since it was of dim R^2, then properties (1) & (2) would be the same. So if it fails (1) & (2) then couldn't you say that it only fails (1)? Response: The 'vectors' are ordered pairs of numbers (hence in R^2), but the 'addition' is defined differently from ordinary addition. The properties (1) and (2) are not the same property. (1) states that the 'addition' (however it is defined) must be commutative and (2) states that it must be associative. It is possible for some 'additions' that (1) holds but (2) doesn't or vice versa.

On 10/20/99, colleter asks about Q17 of MA322-005,08 Section 3.1 and 3.2 (Oct 14): Would you go over pivots, free variables, and echelon form in class. I am having a hard time with these concepts. Response: Sure. I use the term 'free column' and 'free variable' interchangably, so a square matrix could have free variables. These would be the columns that don't contain a pivot. What if I had asked the question about a square invertible matrix? (Answer: true, since each column must contain a pivot in the invertible case.)

On 10/20/99, colleter asks about Q20 of MA322-005,08 Section 3.1 and 3.2 (Oct 14): How can the nullspace that contains only the zero vector have any pivots? Response: Pivots are things that matrices have, not nullspaces, so your question needs to be rethought. (Remember that if a squarematrix is invertible then its nullspace contains only the zero vector and conversely.)

On 10/20/99, colleter asks about Q3 of MA322-005,08 Section 3.1 and 3.2 (Oct 14): I thought that the subspace of a vector had to include the Zero vector? Could you explain your solution to this question in class? Response: This is a typo. The correct alternative should read R^3, not R^2. The smallest subspace of R^3 containing a given plane not through the origin is all of R^3. Thanks for asking about this. I have fixed it now.