Exercise 4. Consider the vector space M22(F) of 2-by-2 matrices with coefficients in F.1) Show that the following subset is a basis of the vector space M22(F)).(11 01 0IConsider the subset V of M22(F) defined by1 -111 1v-{(::)e+d=D0}.|a, b, c, d eR and a + b+ e+d =2) Show that V is a subspace of M22(F) and V # M22(F).3) Find a base of V.4) What is the dimension of the subspace W of M22(F) defined by{(:)..b. c, d e R, b = 0 and a + b+d%3De30?

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Exercise 4. Consider the vector space M22(F) of 2-by-2 matrices with coefficients in F. 1) Show that the following subset is a basis of the vector space M2,2(F) 1 0. -1 Consider the subset V of M22(F) defined by ()1a.bcdeR and a +b+c+d= 0%. V = 2) Show that V is a subspace of M22(F) and V # M22(F). 3) Find a base of V. 4) What is the dimension of the subspace W of M22(F) defined by W = | a,b.c.d e R,b 0 and a + b+d = c

Exercise 4. Consider the vector space M22(F) of 2-by-2 matrices with coefficients in F. 1) Show that the following subset is a basis of the vector space M2,2(F) 1 0. -1 Consider the subset V of M22(F) defined by ()1a.bcdeR and a +b+c+d= 0%. V = 2) Show that V is a subspace of M22(F) and V # M22(F). 3) Find a base of V. 4) What is the dimension of the subspace W of M22(F) defined by W = | a,b.c.d e R,b 0 and a + b+d = c

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.

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Exercise 4. Consider the vector space M22(F) of 2-by-2 matrices with coefficients in F.
1) Show that the following subset is a basis of the vector space M22(F)
).(
1
1 0
1 0
I
Consider the subset V of M22(F) defined by
1 -1
1
1 1
v-{(::)
e+d=D0}.
|a, b, c, d eR and a + b+ e+d =
2) Show that V is a subspace of M22(F) and V # M22(F).
3) Find a base of V.
4) What is the dimension of the subspace W of M22(F) defined by
{(:).
.b. c, d e R, b = 0 and a + b+d%3De30?

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