MTH501 Assignment solution spring 2023 | MTH501 assignment 2023 | MTH501 | vu assignment solution | MTH501 Assignment solution 02 spring 2023
Solution:
Problem Over the field of real numbers R show that the vector space V of all symmetric matrices of order 2 * 2 is 3 dimensional.
Solution:
To show that the vector space of all symmetric matrices of order 2x2 denoted as V is 3-dimensional over the field of real numbers R we need to demonstrate three linearly independent matrices that span V.
Lets consider three matrices
To prove linear independence we need to show that the only solution to the equation c₁A + c₂B + c₃C = 0
(where c₁ c₂ c₃ are scalars) is c₁ = c₂ = c₃ = 0.
Lets set up the equation:
c₁A + c₂B + c₃C = 0
Simplifying the equation we get
=
From this we can see that c₁ = c₂ = c₃ = 0 is the only solution. In this way A B and C are independent linearly.
All three matrices A B and C are symmetric because they are equal to their own transposes.
To show that A B and C span V we need to demonstrate that any symmetric matrix can be expressed as a linear combination of A B and C.
Given an arbitrary symmetric matrix A we can write it as a linear combination of A₁ A₂ and A₃
Which is a linear combination of A₁ A₂ and A₃.
Hence we have shown that the matrices A₁ A₂ and A₃ are linearly independent and span the vector space V of all symmetric matrices of order 2x2 over R. Therefore, V is 3- dimensional.
Since we have found three linearly independent matrices that span V and any additional matrix in V can be expressed as a linear combination of these three matrices we can conclude that the vector space V of all symmetric matrices of order 2x2 over the field of real numbers is indeed 3 dimensional.
Content:
MTH501 Assignment solution spring 2023 | MTH501 assignment 2023 | MTH501 | vu assignment solution | MTH501 Assignment solution 02 spring 2023
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