Solving Basis Problem in R2: Find P Matrix

• snoggerT
In summary, the problem asks for a matrix P such that the R basis vectors (-2,-3) and (-1,-2) can be written as a linear combination of the B basis vectors (4,1) and (1,3). The columns of matrix P will represent the coefficients needed to perform this transformation.
snoggerT
Let B be the basis of R^2 consisting of the vectors :

(4,1) and (1,3)

and let R be the basis consisting of

(-2,-3) and (-1,-2)

Find a matrix P such that [x]_R = P [x]_B for all x in R^2

*note: the vectors are columns, so in (4,1) the 4 is the top row and 1 is the bottom row.

still looking for help on this problem.

How would (4, 1) and (1, 3), the B basis vectors, be written in the R basis?

The columns of matrix P will be those vectors.

Alright, got it. I feel stupid now though.

1. What is the basis problem in R2 and why is it important?

The basis problem in R2 involves finding a set of vectors that can linearly span a two-dimensional vector space. This is important because it allows us to represent any point in the vector space using a combination of these basis vectors, making it easier to solve problems and analyze data.

2. How do you determine the P matrix to solve the basis problem in R2?

The P matrix can be determined by finding a set of linearly independent vectors in the R2 space, which means that none of the vectors can be written as a linear combination of the others. These vectors will form the columns of the P matrix, and the P matrix itself will be used to transform any vector in R2 into a linear combination of these basis vectors.

3. Can the P matrix for solving the basis problem in R2 be unique?

Yes, the P matrix can be unique depending on the choice of basis vectors. However, there can also be multiple P matrices that can solve the basis problem in R2, as long as they have the same set of basis vectors.

4. How do you know if a set of vectors is a basis for R2?

A set of vectors will form a basis for R2 if they are linearly independent and can span the entire two-dimensional vector space. This means that any point in R2 can be written as a linear combination of these basis vectors, and no vector can be written as a combination of the others.

5. Can the basis problem in R2 be extended to higher dimensions?

Yes, the basis problem can be extended to higher dimensions, such as R3 or Rn. The process of finding a set of basis vectors and the P matrix remains the same, but there will be more vectors needed to span the vector space in higher dimensions.

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