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