Showing stuff
What do we have to do to show these things: Show which functions are linear Show that this is a basis Also, can anyone think of anything else we might have to show, and how we are supposed to do it?
justinKennedy: June 4, 2015, 5:37 p.m.
To show a function is linear, you have to show A) T(u + v) = T(u) + T(v) (Closed under addition) B) T(ku) =kT(u) (Closed under scalar multiplication) In doing the problems, you can make u= (u1 u2 u3) and v = (v1 v2 v3) (if transformation has 3 dimensions) and plug them into the transformation to see if (a) and (b) are true. To show something is a basis, you have to show A) Linear Independence; which you can show through c1v1 + c2v2 + c3v3 = 0; if the only way this equation holds true is if c1 = c2 = c3 =0, then its linearly independent. B) Span; in proving span, you're showing that any vector in the space can be written as v=av1 + bv2. so if you set ( x y) = av1 + bv2 and solve for a,b. If they are defined for every x,y , then it spans.
aditya_srinivasan: June 5, 2015, 12:58 a.m.
An easy way to show that something is a valid basis is to write the supposed basis vectors as a matrix and row reduce. If you row reduce and get the identity matrix, you know that the vectors are linearly independent and that they span the entire vector space. If you don't get the identity matrix, then you know that there is some redundancy in the vectors, and therefore, it is not a basis.