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?

- anonymous

- jamiebookeater

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- anonymous

it must at least contain a zero element

- anonymous

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

- anonymous

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?

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- anonymous

usually its obvious

- anonymous

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

- anonymous

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.

- anonymous

you should be able to identify the zero vector

- anonymous

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)

- anonymous

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.

- anonymous

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

- anonymous

o.k

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

Msubn means the set of nxn matrices

- anonymous

yeah.

- anonymous

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

- anonymous

ok.

- anonymous

so whats a lower triangular matrix?

- anonymous

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

- anonymous

yeah.

- anonymous

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

- anonymous

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)

- anonymous

do you mind using twiddla

- anonymous

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

- anonymous

whats that?

- anonymous

http://www.twiddla.com/523796

- anonymous

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.

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