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it must at least contain a zero element
non empty meaning there is at least one element in your vector space. , non empty is pretty self explanatory
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?
usually its obvious
so for square matrices, nxn, the one all zeroes is your zero matrix
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.
you should be able to identify the zero vector
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)
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.
:) why dont you write down a problem youre working on ?
V=Mn(Real), and S is the subset of all nxn lower triangular matrices.
the Mn is, M sub n and real is the real numbers symbol.
Express S in set notation and determine whether it is a subspace of the given vector space V.
Msubn means the set of nxn matrices
so say we have [ 1 1 ] [ 2 2 ] , where [ 1 1 ] is the first row and [ 2 2 ] is the second row
so whats a lower triangular matrix?
it would be [ 1 0 ] [ 1 2 ] for example
a better one would be [ 1 0 0 ] [ 1 2 0 ] [ 1 2 3 ] is a L T matrixx
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)
do you mind using twiddla
so how would you find the zero vector, or better yet how would you know that ur dealing with an nonempty set?
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.