Linear independency before and after Linear Transformation 3 Suppose v1, ⋯vn v 1, ⋯ v n are linearly independent and T is a linear transformation Suppose ker (T) only intersects linear span W W of v1, ⋯vn v 1, ⋯ v n only at {0} {0} Then T preserves the linear independence of v1, ⋯vn v 1, ⋯ v n This condition is necessary and sufficient
Can someone explain these rules about determining linear independence . . . The next two paragraphs explain your second and third point using this If there is a zero column than that column is linearly dependant on the other columns, so no linear independance Same for the third point The two columns that differ by a factor are linearly independant
What does it mean when we say a variable changes linearly? I have attached a screenshot in which a variable is defined for an object somehow that it linearly decreases from 500 micrometers at the top of the object to 50 micrometers at the bottom of the object I was wondering what does it mean by linearly decreases?
Connection between linear independence, non- trivial and x solutions . . . A set of vectors is linearly dependent when there are an infinite amount of solutions to the system of equations This is non-trivial? Where does no solution come in? I understand that if there is no solution, then all of the vectors do not intersect at a specific coordinate (which is the solution to the system of equations)
definition - Is a linear combination linearly independent . . . The vectors are linearly independent if the only linear combination of them that's zero is the one with all αi α i equal to 0 It doesn't make sense to ask if a linear combination of a set of vectors (which is just a single vector) is linearly independent