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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 V of 3x3 given in Exercise 6 through 11 are subspaces of 3x3. The diagonal 3x3 matrices.

The diagonal 3x3 matrices subset V of 3x3 is a subspace of 3x3.

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 3x3 matrix namely,


Find A+B.


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.


Hence, diagonal matrices are a subspace of 3x3.

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