Question

In: Math

Let?:?2(R)⟶?1(R)bedefinedby?(?+?x+?x2)=(?+?)+(?−?)x,where ?, ?, ? are arbitrary constants. a. DeterminethetransformationmatrixforT.(6pts) b. Find the basis and the dimension...

Let?:?2(R)⟶?1(R)bedefinedby?(?+?x+?x²)=(?+?)+(?−?)x,where ?, ?, ? are arbitrary constants.
a. DeterminethetransformationmatrixforT.(6pts)
b. Find the basis and the dimension of the Kernel of T. (10pts)

c. Find the basis and the dimension of the Range of T. (10pts) d. Determine if T is one-to-one. (7pts)
e. DetermineifTisonto.(7pts)

Expert Solution

Solution:

is defined by

a.

Let and be the basis for and

The transformation matrix for is

b.

is the set of all solutions to

Also, is linearly independent.

So, the basis for is

c.

The matrix is in the row reduced echelon form.

In the row reduced echelon form, the first and the third column form the pivot columns .

So, the first and third column form a basis for the range of .

The basis for the is

d.

Since , is not one-to-one.

e.

is onto.

milcah answered 2 years ago

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