Basis Extension Theorem : Linear Algebra : Degree #Basis #Extension #Theorem #: #Linear #Algebra #: #Degree
For a video explanation ,click the link 👉 https://youtu.be/nFliuHhXhRc?si=_Ktb9UG9aDTYf6Lw Basis Extension Theorem : Statement : Let V ( F ) be a finite dimensional vector space and S = { 𝜶 1 , 𝜶 2 , … , 𝜶 n } a linearly independent subset of V. Then S is itself a basis or it can be extended to form a basis of V. 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 of V3 ( F ) * Theorem on Linear Dependence * linear sum of subspaces * What is a vector space * linear span of a set * L ( S ) = L ( S' ) * Theorem on vector space ...