Question

Vector Space:
Let V(2) be the set of all 2 x 3 matrices. Verify that V(2) is a vector space.

Answer 1

your job is to check the axioms for a vector space one at a time

Answer 2

so for example you check that if \(A,B\) are two 2 by 3 matrices, then \(A+B=B+A\) that is, that is commutative. this follows directly from the definition of matrix addition and the fact that real numbers are comx+y = y+x.mutative

Answer 3

what is the multiplicative identity... how do I check if that exists, just multiply by a 3 x 2 matrix full of 1's???

Answer 4

it is an additive identity you are looking for, which is the zero matrix

Answer 5

matrix multiplication is not defined here

Answer 6

vector space axioms have nothing about multiplying two vectors, only scalar multiplication

Answer 7

well i found where it lists the properties of a vector space in my book... and it states.
1. addition is commutative
2. addition is associative
3. additive identity 0 exists
4. additive inverse exists
5. multiplication is associative
6. multiplicative idenity exists
7. distributive law for scalars
8. distributive law for vectors

Answer 8

do i have to prove all 8 just to show that matrix is a vector space

Answer 9

"multiplication is associative" means scalar multiplication

Answer 10

you prove that the set of 2 by 3 matrices form a vector space exactly by showing all three axioms hold. but don't get scalar multiplication confused with matrix multiplication
you cannot multiply two 2 by 3 matrices

Answer 11

yeah, i am understanding all the additive properties, and 5 as well, along with 7 and 8, but i do not know how to show property 6.

Answer 12

again the multiplicative identity means a scalar not a matrix, so it is 1 in this case

Answer 13

just then its just multiplying 1 by the matrix, not an actual matrix full of ones

Answer 14

yes.
all the properties of multiplication are about scalar multiplication
in general you cannot multiply two vectors

Answer 15

here is a list if you need it
http://www.math.ucla.edu/~tao/resource/general/121.1.00s/vector_axioms.html

Answer 16

okay, yes. thank you, i dont know why it was confusing me... it makes sense, this proof is just ridiculous and long. i thank you for your help... i have one more question i am posting if you are going to be around.

Answer 17

it is long, but you will admit that all the steps are utterly trivial. they come from the definitions of matrix addition, and the usual properties of real numbers. the exercise is to get you to learn the axioms, not to prove anything difficult

Answer 18

thank you again. appreciate your help

Answer 19

yw