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 fand gare 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 3x3 matrix namely,

$$\begin{gathered} A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \\ B=\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right] \\ \left|A\right|=1\left(1-0\right)-0+0 \\ \left|A\right|=1 \end{gathered}$$

Find A+B.

$$\begin{gathered} A+B=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right]+\left[\begin{array}{ccc}4& 0& 0\\ 0& -3& 0\\ 0& 0& -1\end{array}\right] \\ A+B=\left[\begin{array}{ccc}4& 0& 0\\ 0& -1& 0\\ 0& 0& 2\end{array}\right] \end{gathered}$$

Thus, the sum of two diagonal matrices is a diagonal matrix, because all other entries will be zero and only the entries in the diagonal can be added and remain non-zero elements.

Similarly, when multiplying a matrix with a scalar, only the non-zero elements in the diagonals can be multiplied and all other will remain as zeros.

$$\begin{gathered} C=2\left[\begin{array}{ccc}4& 0& 0\\ 0& -3& 0\\ 0& 0& -1\end{array}\right] \\ C=\left[\begin{array}{ccc}8& 0& 0\\ 0& -6& 0\\ 0& 0& -2\end{array}\right] \end{gathered}$$

Hence, diagonal matrices are a subspace of ${\mathrm{鈩�}}^{3x3}$.