Help with matrix exercise

[vladek1337]

Hello,

I am having difficulties figuring out a solution to this exercise and I hope you can help me out. I am a first year college student. I attached a photo with the text of the exercise. Any help will be greatly appreciated. Thank you!

 

[stapel]

I am having trouble figuring out a solution to the attached exercise. I hope you can help me solve it. I am a first year college student. Any help will be greatly appreciated. Thank you!

---

\(\displaystyle M\, \subseteq\, M_3(\mathbb{C}),\, M\, \neq\, 0\)

\(\displaystyle \mbox{We know that:}\)

. . . . .\(\displaystyle \mbox{1) }\, A,\, B\, \in\, M\, \Rightarrow\, A\, +\, B\, \in\, M\)

. . . . .\(\displaystyle \mbox{2) }\, A\, \in\, M,\, C\, \in\, M_3\, \Rightarrow\, CA\, \in\, M\)

. . . . .\(\displaystyle \mbox{3) If }\, x\, \in\, M_{3,1}(\mathbb{C})\, \mbox{ and }\, Ax\, =\, 0\, \forall\, A\, \in\, M,\, \mbox{ then }\, x\, =\, 0\)

\(\displaystyle \mbox{Prove that }\, M\, =\, M_3(\mathbb{C}).\)

I'm assuming that the first C stands for "the complex numbers". But what are M, M_3, M_(3,1), A, and the C in part (2)?

When you reply, please include a clear listing of your thoughts and efforts so far. Thank you!

 

[vladek1337]

That's correct, the first C stands for the complex numbers
M_3 is the set that is including all 3x3 matrix
M is a set included in M_3
M_(3,1) is a set which includes all the 3x1 matrix
A and C are matrix

Unfortunately, I am having big trouble understanding the way I should start my demonstration as I am barely seeing a connection between 1,2 and 3, because I didn't really encounter a similar exercise until this one. I would like, if possible, to give me the solution in order to understand what should I have in mind when dealing with exercises like this. Thank you for your help and patience!

 

[Steven G]

Hello,

I am having difficulties figuring out a solution to this exercise and I hope you can help me out. I am a first year college student. I attached a photo with the text of the exercise. Any help will be greatly appreciated. Thank you!

You are given that M is in M3(C) and want to show that M=M3(C). So you need to show that M3(C) is in M.
Try to show this. Please respond with your work. If it is correct, that will be great and we will give you another hint. If you have a mistake with your work we will help you out seeing your error.