Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Which of the subsets $V$ of $ℝ^{3 \times 3}$ given in Exercise through 11 are subspaces of $ℝ^{3 \times 3}$ . The role="math" localid="1659358236480" $3 x 3$ 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 $ℝ^{3 \times 3}$ is not a subspace of $ℝ^{3 \times 3}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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 $3 \times 3$ matrices with entries positive or zero. Let,

$A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}]$

$B = [\begin{matrix} 1 & 4 & 8 \\ 9 & 5 & 1 \\ 1 & 2 & 3 \end{matrix}]$

Find. $A + B$

$A + B = [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}] + [\begin{matrix} 1 & 4 & 8 \\ 9 & 5 & 1 \\ 1 & 2 & 3 \end{matrix}] = [\begin{matrix} 2 & 4 & 8 \\ 9 & 7 & 1 \\ 1 & 2 & 6 \end{matrix}]$

Thus, it is closed under addition.

Multiply the matrix by any scalar,

$A = - 1 [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}]$

$= [\begin{matrix} - 1 & 0 & 0 \\ 0 & - 2 & 0 \\ 0 & 0 & - 3 \end{matrix}]$

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 $ℝ^{3 \times 3}$ .

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Linear Algebra With Applications

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Short Answer

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

Step by Step Solution

Step 1: Definition of subspace.

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

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Linear Algebra With Applications

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Short Answer

Which of the subsets Vof ℝ3×3 given in Exercise through 11 are subspaces of ℝ3×3. The role="math" localid="1659358236480" 3x3matrices whose entries are all greater than or equal to zero.

Step by Step Solution

Step 1: Definition of subspace.

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

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Pure Maths

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