Why is 2x2 matrix [a, 1; 1,b] not at vector space, but [a,0; 0,b] a vector space?

Question

vector:
[a 1]
[1 b]

vs. vector
[a 0]
[0 b]

Answer 1

Well the first one does not have the 0 indentity. Therefore, not a vector space.



The second contains the 0 identity, is closed under addition and is closed under scalar multiplication. Therefore, its a vector space.



Note to verify if something is a vector space you must do the following:



1) Is 0 in the vector space?

2) If x and y are in the vector space is x + y in the vector space. (This is called closed under addition)

3) If x is in the vector space and c is any scalar then cx is in the vector space. (This is called closed under scalar multiplication)



If these are all true then its a vector space!



Hope this helps!

Answer 2

1) correct. 2) correct. 3) no. if M is invertible, with inverse N, then (-M)^-1 = -N, so -M is invertible. 4) correct. 5) correct, the 0-matrix is not in the set, so it has no additive identity. 6) yes. but if 1) doesn't hold, this isn't too useful. 7) sure, the operation is the same as in the set of all 2x2 matrices. 8) no, this axiom still holds. invertible matrices still have this property, it's just that the results may not be invertible matrices. 9) see 8 above. 10). again, see 8, and 9. the crucial failures are closure of vector addition and scalar multiplication. this means that they are not well-defined operations on the set of invertible matrices.

Answer 3

It's because any vector space needs to be closed under certain operations of scalar multiplication and addition.



For the first vector, if we scalar multiply by 2, we'll get

[2a, 2]

[2, 2b]

which doesn't fit the form of the vector given (we have 2's where you had 1's)

Answer 4

[a 0]

[0 b]

is the general form for a vector space because a vector space is ab.

determinant of first matrix is ab-1.

determinant of second matrix is ab.

so its the second one.

Answer 5

you mean the set of those matrices

"why is the set of matrices of the form [a,1;1,b] not a vector space over R, but the set of matrices of the form [a,0; 0,b] is a vector space over R?"



answer: the first does not have a zero element

while the second has a zero element, namely [0,0; 0,0]