Since the coordinate mapping determined by B is a linear

Chapter 4, Problem 21E

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QUESTION:

Problem 21E

Since the coordinate mapping determined byB is a linear transformation from  into , this mapping must be implemented by some 2 × 2 matrix A. Find it. [Hint: Multiplication by A should transform a vector x into its coordinate vector

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QUESTION:

Problem 21E

Since the coordinate mapping determined byB is a linear transformation from  into , this mapping must be implemented by some 2 × 2 matrix A. Find it. [Hint: Multiplication by A should transform a vector x into its coordinate vector

ANSWER:

Solution  21E

Step 1

Let us suppose that,

Observe that,  where

Observe that, the two vectors in are not scalar multiples each other, so these two vectors are linearly independent in and it has dimension 2. So the set  forms a basis for

Let

The coordinates of relative to the basis  are the weights  such that

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