Rank, Dimension, Subsapce, Column Space

[kingwinner]

1) True or False? If true, prove it. If false, prove that it is false or give a counterexample.
1a) If A is m x n, then A and (A^T)(A) have the same rank.
1b) Let A be m x n and X E R^n. If X E null [(A^T)(A)], then AX is in both col(A) and null(A^T).
[I believe it's true that AX is in null(A^T), but I am totally unsure whether AX is in col(A) or not!?]
1c) If U and W are subspaces of a vector space V, then the set of vectors that belong to either U or W is a subspace of V.



2) Prove that if A is an m x n matrix, then null(A)=[col(A^T)]^|
[Using dimension theorem, I proved that their dimensions are equal...but I have no idea how to prove that they ARE equal...]



These are also the past exams questions that I am having terrible trouble with. Can someone give me some advice/hints? For 1b) and 2), I am partially done, but how 1a)c) I have no clue...

Any help/hints is greatly appreciated!

 

[radou]

1a) Have you proved that the column rank equals the row rank? Then the proof of of 1a is trivial.
1c) First of all, if you have two sets, what set operation fits into what you need, i.e. "the set of vectors that belong either to U or to W"?

 

[siddharth]

1c) If U and W are subspaces of a vector space V, then the set of vectors that belong to either U or W is a subspace of V.

Are you sure that's the entire question? If $$U$$ and $$W$$ are subspaces of a vector space $$V$$, then $$U \cup W$$ is a subspace if and only if $$U \subseteq W$$ or $$W \subseteq U$$

2) Prove that if A is an m x n matrix, then null(A)=[col(A^T)]^|
[Using dimension theorem, I proved that their dimensions are equal...but I have no idea how to prove that they ARE equal...]

Now, the column space of A^T is the row space of A. So, you need to show that
(i) the nullspace of A is a subset of [row(A)]^| . ie, show that every element which belongs to the null space also belongs to [row(A)]^|.
(ii) [row(A)]^| is a subset of the nullspace of A. ie, every element u in [row(A)]^| is also in the null space.

 

[kingwinner]

1a) Have you proved that the column rank equals the row rank? Then the proof of of 1a is trivial.
1c) First of all, if you have two sets, what set operation fits into what you need, i.e. "the set of vectors that belong either to U or to W"?

1a) I have learned that rank A=dim(colA)=dim(rowA), but how does that help?

1c) The union space is larger, so my guess is that it would still be a subspace, right?

 

[kingwinner]

Are you sure that's the entire question? If $$U$$ and $$W$$ are subspaces of a vector space $$V$$, then $$U \cup W$$ is a subspace if and only if $$U \subseteq W$$ or $$W \subseteq U$$



Now, the column space of A^T is the row space of A. So, you need to show that
(i) the nullspace of A is a subset of [row(A)]^| . ie, show that every element which belongs to the null space also belongs to [row(A)]^|.
(ii) [row(A)]^| is a subset of the nullspace of A. ie, every element u in [row(A)]^| is also in the null space.

1c) Yes, so I guess the answer is "false". But why? I don't understand...


2) But how can I relate null A to (rowA)^|? I can think of no way of showing them to be equal...

Thanks!

 

[kingwinner]

1b) How can I know whether AX is in col(A) or not?