Some ridiculously tough true and false questions (Linear Algebra)

[flyingpig]

Homework Statement

http://img844.imageshack.us/img844/8878/truie.th.png

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The Attempt at a Solution

a) I know that it contains the 0 vector, so one of the conditions for subspace. But how do I determine whether the plane contains the origin?

b) I am going to answer False, because Ax = b is still the columnspace (provided b is not 0). This is only true if b = 0 (which is the nullspace)

c) I honestly do not even know where to begin lol. The addition part is killing me

d) Is this too trivial? Does this fall from Diagonalization? I am going to say it is true. I don't have a formal proof though...

 

[Mark44]

Homework Statement

http://img844.imageshack.us/img844/8878/truie.th.png

Uploaded with ImageShack.us

The Attempt at a Solution

a) I know that it contains the 0 vector, so one of the conditions for subspace. But how do I determine whether the plane contains the origin?

What are the coordinates of the origin? How would you tell if any given point is on the plane?

b) I am going to answer False, because Ax = b is still the columnspace (provided b is not 0). This is only true if b = 0 (which is the nullspace)

c) I honestly do not even know where to begin lol. The addition part is killing me

Isn't A² + 5A = A(A + 5I)?

d) Is this too trivial? Does this fall from Diagonalization? I am going to say it is true. I don't have a formal proof though...

 

[flyingpig]

a)What are the coordinates of the origin?

The coord of origin is <0,0,0>

How would you tell if any given point is on the plane?

How about 0 + 2(0) + 3(0) = 0 ≠ 4, so it doesn't contain the origin. Oh it is false then...

 

[flyingpig]

Isn't A2 + 5A = A(A + 5I)?

I swear, I did NOT see the 5A when I did the problem

So det(A(A + 5I)) = 0 = det(A)det(A + 5I) = 0 * det(A + 5I) = 0 = 0

So it is true!

 

[Mark44]

How about 0 + 2(0) + 3(0) = 0 ≠ 4, so it doesn't contain the origin. Oh it is false then...

Yes.

I swear, I did NOT see the 5A when I did the problem

So det(A(A + 5I)) = 0 = det(A)det(A + 5I) = 0 * det(A + 5I) = 0 = 0

So it is true!

Yes.