9. A vector space V is called the direct sum of W1 and W2 if W1 and W2 are subspaces of V such that W1W2 = {0} and W1+W2 = V. We denote that V is the direct sum W1 and W2 by writing V = W1 W2. Show that the space M2 (R) of 2 × 2 matrices with real entries is the direct sum of two subspaces M2 (R) = W1 ☺ W2, where - {[ : :), abe3}, w.= {[i <], adez}. a b W1 = W2

9. A vector space V is called the direct sum of W1 and W2 if W1 and W2 are subspaces of V such that W1W2 = {0} and W1+W2 = V. We denote that V is the direct sum W1 and W2 by writing V = W1 W2. Show that the space M2 (R) of 2 × 2 matrices with real entries is the direct sum of two subspaces M2 (R) = W1 ☺ W2, where - {[ : :), abe3}, w.= {[i <], adez}. a b W1 = W2

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

9. A vector space V is called the direct sum of W1 and W2 if W1 and W2 are subspaces
of V such that W1NW2 = {0} and W1 + W2 = V. We denote that V is the direct sum
W1 and W2 by writing V = W1 W2. Show that the space M2 (R) of 2 × 2 matrices
with real entries is the direct sum of two subspaces M2 (R) = W1 O W2,
where
{[: 4], adez}.
a
C
Wi =
а, bE R
W2
-6
a

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