Today we further discuss Remark 1 in the last post – what happens when we perform set operations on two subspaces. A key point is that we are led to the concepts of *the sum of two subspaces*, and the *direct sum*. (The lecture slides are uploaded in the folder Slides at moodle.)

After the lecture ended, some classmates asked questions that I would share with you here.

- Qn 1. Is the direct sum (of subspaces) related to quotient vector spaces?
You may wonder why your classmate ask this question because they (direct sum and quotient space) seem to be very different objects. If the vector space , then and are containing vectors in . However, contains the equivalence classes where , which is more abstract and complicated. Nevertheless, there are interesting relations between the concepts of direct sum and quotient vector space. You will get some ideas in Tutorial 2 and Assignment 2.

- We explained in lecture that even if and are subspaces of , is NOT a subspace of . Our proof is that both and contain the zero vector ; thus and so is not a subspace.
Qn 2. A classmate asked why the zero vector of must equal the zero vector of .

(Well our proof does not work if they are different.) This question sounds silly but indeed it makes sense very much! If you were me, how would you reply?

Mine is: the zero vector of any subspace (of ) has to be the zero vector of .

First of all,

*zero vector in a vector space is unique*(i.e. given a vector space, there is only one element fulfilling (A4)). [Prove it! If you cannot work it out, please ask me or our tutor.]Next, let be a vector space and be its subspace. Suppose is the zero vector of and is the zero vector of . Now , i.e. is a vector in . Write for the real number zero and use Part (1) of Theorem 6.1.3 (i.e. Thm 3 of Section 6.1) twice, we infer

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I guess there is a typo in the ‘mine is’ line.

Comment by Shum Ho Pan — September 16, 2014 @ 12:58 AM |

Thanks. Amended.

Comment by Anonymous — September 16, 2014 @ 10:23 AM |