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 u1u_1 and u2u_2 in R2\mathbb{R}^2 such that the vector s=[6,13,1,2]s = [6, -13, 1, -2] represents the change-of-basis matrix from the basis B={v1,v2}B = \{ v_1, v_2 \}, where v1=[13,8]v_1 = [13, -8] and v2=[15,11]v_2 = [15, -11], to the new basis C={u1,u2}C = \{ u_1, u_2 \}.

Approach

We are given a vector ss that is structured as [6,13,1,2][6, -13, 1, -2], and this corresponds to the transformation matrix from BB to CC. This matrix TT is:

T=(61132)T = \begin{pmatrix} 6 & 1 \\ -13 & -2 \end{pmatrix}

This matrix transforms the basis B={v1,v2}B = \{ v_1, v_2 \} into the new basis C={u1,u2}C = \{ u_1, u_2 \}.

Step-by-Step Solution

  1. Matrix Representation: Let TT be the transformation matrix that takes vectors from basis BB to basis CC. Then:

    (u1u2)=T(v1v2)\begin{pmatrix} u_1 \\ u_2 \end{pmatrix} = T \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}

    In this case:

    T=(61132),v1=(138),v2=(1511)T = \begin{pmatrix} 6 & 1 \\ -13 & -2 \end{pmatrix}, \quad v_1 = \begin{pmatrix} 13 \\ -8 \end{pmatrix}, \quad v_2 = \begin{pmatrix} 15 \\ -11 \end{pmatrix}

  2. Find u1u_1: To find u1u_1, multiply the transformation matrix TT by the vector v1v_1:

    u1=Tv1=(61132)(138)u_1 = T \cdot v_1 = \begin{pmatrix} 6 & 1 \\ -13 & -2 \end{pmatrix} \begin{pmatrix} 13 \\ -8 \end{pmatrix}

    Perform the matrix-vector multiplication:

    u1=((6×13)+(1×8)(13×13)+(2×8))u_1 = \begin{pmatrix} (6 \times 13) + (1 \times -8) \\ (-13 \times 13) + (-2 \times -8) \end{pmatrix}

    Simplifying:

    u1=(788169+16)=(70153)u_1 = \begin{pmatrix} 78 - 8 \\ -169 + 16 \end{pmatrix} = \begin{pmatrix} 70 \\ -153 \end{pmatrix}

  3. Find u2u_2: Similarly, to find u2u_2, multiply the transformation matrix TT by the vector v2v_2:

    u2=Tv2=(61132)(1511)u_2 = T \cdot v_2 = \begin{pmatrix} 6 & 1 \\ -13 & -2 \end{pmatrix} \begin{pmatrix} 15 \\ -11 \end{pmatrix}

    Perform the matrix-vector multiplication:

    u2=((6×15)+(1×11)(13×15)+(2×11))u_2 = \begin{pmatrix} (6 \times 15) + (1 \times -11) \\ (-13 \times 15) + (-2 \times -11) \end{pmatrix}

    Simplifying:

    u2=(9011195+22)=(79173)u_2 = \begin{pmatrix} 90 - 11 \\ -195 + 22 \end{pmatrix} = \begin{pmatrix} 79 \\ -173 \end{pmatrix}

Final Answer

The vectors u1u_1 and u2u_2 are:

u1=(70153),u2=(79173)u_1 = \begin{pmatrix} 70 \\ -153 \end{pmatrix}, \quad u_2 = \begin{pmatrix} 79 \\ -173 \end{pmatrix}

These are the vectors in the new basis CC that correspond to the change of basis from B={v1,v2}B = \{ v_1, v_2 \} to C={u1,u2}C = \{ u_1, u_2 \}.


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:

  1. How do we interpret the geometric meaning of a change of basis in R2\mathbb{R}^2?
  2. Can the transformation matrix TT be interpreted as a linear map between vector spaces?
  3. What happens if the matrix TT is not invertible?
  4. How can we find the inverse of a change-of-basis matrix?
  5. How would this problem change if the vectors v1v_1 and v2v_2 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