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Linear Algebra With Applications
Found in: Page 176
Linear Algebra With Applications

Linear Algebra With Applications

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

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Short Answer

Which of the subsets Vof 3×3 given in Exercise through 11 are subspaces of 3×3. The role="math" localid="1659358236480" 3x3matrices 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 3×3 is not a subspace of 3×3.

See the step by step solution

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 3×3matrices with entries positive or zero. Let,




A+B=100020003+148951123 =248971126

Thus, it is closed under addition.

Multiply the matrix by any scalar,



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 3×3 .

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