Mahatma jyotiba phule Linear Algebra.pptx

Linear Algebra

1. Subject : Linear Algebra Topic : Linear Sum of Two Subspaces Department of Applied Mathematics (2023-25) Submitted By: Anjali Sharma

2. Definition – Let V(F) be a vector space and + = { , Definition – Let V(F) be a vector space and subspaces is denoted by and is defined as- + = { ,   LINEAR SUM OF TWO SUBSPACES

3. Theorem- Statement : The linear sum of two subspaces of vector space V is the smallest subspace of containing both . Proof : Given that, V is a vector space and be its two subspaces then the linear sum of is defined as- = { , We have to show that is the smallest subspace of V containing both . First, we will show that is a subspace of V. Thus, is non-empty subset of V.  

4. Let, + , + , and , + and F {and are subspaces } Now, F, we have ) + ) = + ++ = ++ + where, + + Thus, ++ + ) + ) Hence, is a subspace of V. Let, + , + , and , + and F { and are subspaces } Now, F, we have ) + ) = + + + = + + + where, + + Thus, + + + ) + ) Hence, is a subspace of V.  

5. Now, we will show that Let, Then, + Thus, Let, We know that, + Thus, Hence, V is the smallest subspace of V containing both and . Now, we will show that Let, Then, + Thus, Let, We know that, + Thus, Hence, V is the smallest subspace of V containing both and .  

6. Hence, and are contained in + . Let, S be an arbitrary subspace of V containing both and . Let,+ , But, S , S + S ( S is a subspace ) Thus, Since, S is arbitrary therefore is the smallest subspace of V containing both and . Hence, and are contained in + . Let, S be an arbitrary subspace of V containing both and . Let, + , But, S , S + S ( S is a subspace ) Thus, Since, S is arbitrary therefore is the smallest subspace of V containing both and .