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)

Solutions

Expert Solution

Solution:

is defined by

Let and be the basis for and

The transformation matrix for is

is the set of all solutions to

Also, is linearly independent.

So, the basis for is

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

Since , is not one-to-one.

is onto.

milcah answered 1 year ago

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