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Q9E

Expert-verifiedFound in: Page 176

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Which of the subsets **${\mathbf{}}{\mathbf{V}}$**of **** ${{\mathbf{\mathbb{R}}}}^{\mathbf{3}\mathbf{\times}\mathbf{3}}{\mathbf{}}$given in Exercise ****through 11**** are subspaces of ${{\mathbf{\mathbb{R}}}}^{\mathbf{3}\mathbf{\times}\mathbf{3}}$****. The ****role="math" localid="1659358236480" ${\mathbf{3}}{\mathbf{x}}{\mathbf{3}}$matrices whose entries are all greater than or equal to zero.**

The $3\times 3$ matrices whose entries are all greater than or equal to zero of ${\mathrm{\mathbb{R}}}^{3\times 3}$ is not a subspace of ${\mathrm{\mathbb{R}}}^{3\times 3}$.

**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.

Consider two $3\times 3$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.$A+B$

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

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,$3\times 3$ a matrix with entries greater than or equal to zero is not a subspace of ${\mathrm{\mathbb{R}}}^{3\times 3}$ .

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