Week 12 Flashcards
(19 cards)
How to show a function is a linear transformation?
T(u+v)=T(u)+T(v)
T(cu)=c⋅T(u)
Notation for a reflection in x axis?
T(e1) T(e2) = (1 0 )
(0 -1)
Notation for a reflection in y=x?
T(e1) T(e2) = ( 0 1)
( 1 0)
Matrix Notation for a stretch 2 in e1 and 3 in e2?
At = (T(e1) T(e2)) = ( 2 0)
( 0 3)
How to find matrix A t B->B ?
At B->B = (T(f1)B T(f2)B)
First we calc T(f1) then do it with respect to basis
So T(f1) = a(f1)+b(f2) then simultaneous to calc T(f1)B
where f1,f2 is basis of B
How to find matrix A t B->C?
At B->C = (T(f1)C T(f2)C)
First we calc T(f1) then do it with respect to basis
So T(f1) = a(g1)+b(g2) then simultaneous to calc T(f1)C
where f1,f2 is basis of B
and g1,g2 is basis of C
How to get coordinate vector (X)b
X = a(f1) +b(f2) , we can rearrange for a and b
How to get coordinate vector (T(x))b
T(x)= a(f1)+b(f2) , then just solve for a and b
T(x)c eq?
T(x)c = At B->C (X)b
Finding At B->C with function?
Instead of T(f1) it’s T(f1)(x)
How to find range of T (change of basis matrix)
Use CS and multiply by new basis , then f(x) = scalar x (what we get) (if multiple then makeup of them)
check in form f(x) = question gives
How to find kernel of T (change of basis matrix)?
Use NS and multiply by original basis , then f(x) = scalar x (what we get)
check in form f(x) = question gives
For kernel and range BASIS?
then we use CS and N
When matrix is invertable A=I what is kernel?
zero function or { }
When asked to find kernel and range just given by cartesian requirements?
Use dimensions from eq to get number of vectors for basis
So we use these no of free parameters for kernel or range
2 ways of calc cartesian of basis of null?
Eliminating t if 1 vector, if loads we can just use the row space basis
Ways of getting cartesian from cs basis?
Just add all the free parameter eq to make 1 vector and solve
if 2 vectors just work out normal using cross product
How to show that any linear transformation must map the zero vector in V to
the zero vector in W.
Use that T(0) = T(0x) = 0T(X) = 0
Ways of working out what a linear transformation looks like?
Use matrix on the standard ordered basis
