Question

Subspace question: So in order for a subset S in vector space V to be a subspace it must have a nonempty set, closed under addition and scalar multiplication. What do they mean by nonempty set? How do you prove that it is nonempty?

Answer 1

it must at least contain a zero element

Answer 2

non empty meaning there is at least one element in your vector space. , non empty is pretty self explanatory

Answer 3

Yeah, i see that in most problems when they have an equation and its set equal to zero. Is it the same when they give you a MxN matrix? how do you know that it has the zero vector?

Answer 4

usually its obvious

Answer 5

so for square matrices, nxn, the one all zeroes is your zero matrix

Answer 6

For the most part I just assume that the subset is not empty and just rely on proving whether its closed under addition and scalar multiplication.

Answer 7

you should be able to identify the zero vector

Answer 8

it is important that you find or identify the zero vector when you are establishing a subspace, (a subset of your original vector space which is also a vector space in its own right)

Answer 9

yeah, I know that my method of approaching these are wrong. I guess I am just thinking too much into it. Thanks for the help.

Answer 10

:) why dont you write down a problem youre working on ?

Answer 11

o.k

Answer 12

V=Mn(Real), and S is the subset of all nxn lower triangular matrices.

Answer 13

the Mn is, M sub n and real is the real numbers symbol.

Answer 14

Express S in set notation and determine whether it is a subspace of the given vector space V.

Answer 15

Msubn means the set of nxn matrices

Answer 16

yeah.

Answer 17

so say we have [ 1 1 ] [ 2 2 ] , where [ 1 1 ] is the first row and
[ 2 2 ] is the second row

Answer 18

ok.

Answer 19

so whats a lower triangular matrix?

Answer 20

it would be [ 1 0 ] [ 1 2 ] for example

Answer 21

yeah.

Answer 22

a better one would be [ 1 0 0 ] [ 1 2 0 ] [ 1 2 3 ] is a L T matrixx

Answer 23

ok there are two conditions for a subspace , since we inherit the vector space properties from the fact that we are in the same set (subset)

Answer 24

do you mind using twiddla

Answer 25

so how would you find the zero vector, or better yet how would you know that ur dealing with an nonempty set?

Answer 26

whats that?

Answer 27

http://www.twiddla.com/523796

Answer 28

looool a lower triangular matrix is a matrix with all zeros below the major diagonal, whereas an upper triangular matrix has all zeros above the major diagonal. this definition might be a bit easier to see. One thing that is neat about these is that all you need to do to take the determinant of a matrix like this is multiply all numbers in the major diagonal. So if you had a test and you noticed the determinant of a matrix as mentioned by cantorset you could do that and you would spend a lot less time.