Plz give me the idea for answer this question.
Let W be the set of matrices of $\displaystyle \begin{bmatrix}
0 & a \\
b & c \\
d & e
\end{bmatrix}
$. Is this a subspace of $\displaystyle M_{32}$
Hi thanks for the reply.
Here you mean about is this matrix
$\displaystyle
$$\displaystyle \begin{bmatrix}
0 & a \\
b & c \\
d & e
\end{bmatrix}
$,
is might be a vector space over K with respect to the usual operations of matrix addition and scalar multiplication?
Here I know if $\displaystyle u=(a_1, a_2, a_3, ...a_n)$ then this multiple by any sclar k we obtain $\displaystyle ku=(ka_1, ka_2, ka_3, .... ka_n)$, so then if this is a vector space need to prove k0=0, 0u=0, ku=0 or (-k)u=k(-u)=-ku isn't it?
now I need to know if I do
$\displaystyle k
$$\displaystyle \begin{bmatrix}
0 & a \\
b & c \\
d & e
\end{bmatrix}
$ then I can't get this is 0. So where is the incorrect place/method and how I go on... pls explain me.
Also this state $\displaystyle M_{32}$ this mean the element "e" or what?
I'm kind of confused by your answer, but it is simple to check and you've indicated you know hot to check it: it is closed, and does it contain the zero vector? ALL other of the ten or so "checks" follow from these two conditions, so there isn't a need to check all of them (if one of the "minor" checks fails, then closure will fail).
A matrix is in this set if and only if its "upper left" entry is 0.
Is it closed under addition? That is, if two matrices both have there "upper left" entry 0, does their sum have the same property?
Is it closed under scalar multiplication? That is, if a matrix has its "upper left" entry 0, does its product by any number have the same property?
Does it include the 0 matrix?
(By the way, this last is equivalent to "is the set non empty?" If you have already proved that a set of vectors is "closed under addition" and "closed under scalar multiplication", then as long as the set contains a single vector, v, it must also contain v+ (-1)v= 0.)