V/W FORMS A VECTOR SPACE : LINEAR ALGEBRA : B.SC. MATHEMATICS #linear algebra #degree
For a video explanation, click the link 👉 https://youtu.be/unnIkQ7DhuM THEOREM : Let W be a subspace of a vector space V ( F ) . Then the set V/W = { W + 𝛂 / 𝛂 𝞊 V } of all cosets forms a Vector Space under the Coset addition and Scalar multiplication are defined as i. For W + 𝛂 , W + 𝞫 𝞊 V, ( W + 𝛂 ) + ( W + 𝞫 ) = W + ( 𝛂 + 𝞫 ) ii . For a 𝞊 F , W + 𝛂 𝞊 V, a ( W + 𝛂 ) = ( W + a𝛂 ). PROOF : * Find the coordinates of 𝜶 = (4 ,5 ,6) w.r.t. x=(1,1,1),y = (-1,1,1),z = (1,0,-1) *dim ( W1 + W2 ) = dim ( W1 ) + dim ( W2) - dim ( W1 ⋂ W2 ) * If 𝜶 = ( 1 , -1 , 0 ) , 𝜷 = ( 2 , 1 , 3 ) find a basis for R3 containing 𝜶 and 𝜷 * Show that the set { ( 1 , 2 , 1 ) , ( 2 , 1 , 0 ) , ( 1 , -1 , 2 } forms a basis o...