91影视

A subset Wof a linear space Vis called a subspace of Vif

(a) W contains the neutral element 0of V.

(b) W is closed under addition (if fandg are in Wthen so is f+g)

(c) W is closed under scalar multiplication (if fis in Wand kis scalar, then kfis in W).

we can summarize parts b and c by saying that W is closed under linear combinations.

Consider two $3脳3$matrices with entries positive or zero. Let,

$A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right]$

$B=\left[\begin{array}{ccc}1& 4& 8\\ 9& 5& 1\\ 1& 2& 3\end{array}\right]$

Find.$A+B$

$$\begin{gathered} A+B=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right]+\left[\begin{array}{ccc}1& 4& 8\\ 9& 5& 1\\ 1& 2& 3\end{array}\right] \\ =\left[\begin{array}{ccc}2& 4& 8\\ 9& 7& 1\\ 1& 2& 6\end{array}\right] \end{gathered}$$

Thus, it is closed under addition.

Multiply the matrix by any scalar,

$A=-1\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right]$

$=\left[\begin{array}{ccc}-1& 0& 0\\ 0& -2& 0\\ 0& 0& -3\end{array}\right]$

As there are negative entries with multiplied with a scalar, this matrix is not closed under scalar multiplication.

Hence,$3脳3$ a matrix with entries greater than or equal to zero is not a subspace of${\mathrm{鈩�}}^{3脳3}$ .