Math Problem Statement
let v1=[13,-8], V2=[15,-11] s=[6,-13,1,-2] find vectors u1, u2 in r2 such that s is the chance of basis matrix from B={v1,v2} to basis C={u1,u2}
Solution
We are tasked with finding the vectors and in such that the vector represents the change-of-basis matrix from the basis , where and , to the new basis .
Approach
We are given a vector that is structured as , and this corresponds to the transformation matrix from to . This matrix is:
This matrix transforms the basis into the new basis .
Step-by-Step Solution
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Matrix Representation: Let be the transformation matrix that takes vectors from basis to basis . Then:
In this case:
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Find : To find , multiply the transformation matrix by the vector :
Perform the matrix-vector multiplication:
Simplifying:
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Find : Similarly, to find , multiply the transformation matrix by the vector :
Perform the matrix-vector multiplication:
Simplifying:
Final Answer
The vectors and are:
These are the vectors in the new basis that correspond to the change of basis from to .
Would you like further clarification on this process, or do you have any specific questions?
Here are 5 related questions to expand on this concept:
- How do we interpret the geometric meaning of a change of basis in ?
- Can the transformation matrix be interpreted as a linear map between vector spaces?
- What happens if the matrix is not invertible?
- How can we find the inverse of a change-of-basis matrix?
- How would this problem change if the vectors and were linearly dependent?
Tip: When performing matrix multiplication, always verify the dimensions of the matrices to ensure the operation is valid.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Change of Basis
Matrix Transformations
Vector Spaces
Formulas
Change of Basis Formula: u = T ⋅ v
Matrix-vector multiplication
Theorems
Matrix Multiplication Theorem
Basis Transformation Theorem
Suitable Grade Level
Undergraduate Level