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Found in: Page 176

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Which of the subsets V of ${{\mathbf{ℝ}}}^{\mathbf{3}\mathbf{x}\mathbf{3}}$ given in Exercise 6 through 11 are subspaces of ${{\mathbf{ℝ}}}^{\mathbf{3}\mathbf{x}\mathbf{3}}$. The diagonal 3x3 matrices.

The diagonal 3x3 matrices subset V of ${\mathrm{ℝ}}^{3x3}$ is a subspace of ${\mathrm{ℝ}}^{3x3}$.

See the step by step solution

## Step 1: Definition of subspace

A subset W of a linear space V is called a subspace of V if

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

(b) W is closed under addition (if f and g are in W then so is f+g )

(c) W is closed under scalar multiplication (if f is in W and k is scalar, then kf is in W).

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

## Step 2: Verification whether the subset is closed under addition and scalar multiplication.

Consider two 3x3 matrix namely,

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

Find A+B.

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

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.

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

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