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Q9E

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

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

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

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