Let B1={ (1,2), (2,-1)} and B2 = {(1,0), (0,1)}  be ordered bases of R^2. If T: R^2 -> R^2 is a linear transformation such that [A] is transition matrix from B1 to B2 is given below. Then find T(5,5) ? where [A]_{B1}^{B2}=[4    3; 2    -4]
1 point
(-9, 8)
(9, 8)
(-15,  -2)
(15,  2)

Given:
- \( B1 = \{(1, 2), (2, -1)\} \)
- \( B2 = \{(1, 0), (0, 1)\} \)
- The transition matrix \([A]_{B1}^{B2}\) is \(\begin{pmatrix} 4 & 3 \\ 2 & -4 \end{pmatrix}\)
- We need to find \( T(5, 5) \)

### Step 1: Find the coordinates of \((5, 5)\) in \( B1 \)

Let \((5, 5)\) in coordinates with respect to \( B1 \) be \((a, b)\), i.e., \( 5(1, 0) + 5(0, 1) \).

So, we need to solve:
\[ a(1, 2) + b(2, -1) = (5, 5) \]
Which translates to:
\[ \begin{cases} a + 2b = 5 \\ 2a - b = 5 \end{cases} \]

Solve this system of equations:
1. From the first equation: \( a + 2b = 5 \)
2. From the second equation: \( 2a - b = 5 \)

Multiply the first equation by 1 and second equation by 2:
\[ \begin{cases} a + 2b = 5 \\ 4a - 2b = 10 \end{cases} \]

Add the equations:
\[ 5a = 15 \implies a = 3 \]

Substitute \( a = 3 \) into \( a + 2b = 5 \):
\[ 3 + 2b = 5 \implies 2b = 2 \implies b = 1 \]

Therefore, \((5, 5)\) in \( B1 \) coordinates is \((3, 1)\).

### Step 2: Apply the linear transformation \( T \)

The given matrix \([A]_{B1}^{B2}\) is:
\[ [A]_{B1}^{B2} = \begin{pmatrix} 4 & 3 \\ 2 & -4 \end{pmatrix} \]

We need to multiply this matrix by the vector \((3, 1)\):
\[ T_{B2} (3, 1) = \begin{pmatrix} 4 & 3 \\ 2 & -4 \end{pmatrix} \begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} 4 \cdot 3 + 3 \cdot 1 \\ 2 \cdot 3 + (-4) \cdot 1 \end{pmatrix} = \begin{pmatrix} 12 + 3 \\ 6 - 4 \end{pmatrix} = \begin{pmatrix} 15 \\ 2 \end{pmatrix} \]

### Step 3: Find \( T(5, 5) \) in standard basis \( B2 \)

The coordinates \((15, 2)\) are already in standard basis \( B2 \).

So, \( T(5, 5) = (15, 2) \).

### Final Answer

\[
\boxed{(15, 2)}
\]

Do you want further details or have any questions about this problem? Here are some related questions for you:

1. How do you find the coordinates of a vector in a given basis?
2. What is the importance of transition matrices in linear algebra?
3. How can you find the inverse of a transition matrix?
4. How do you apply a linear transformation using a matrix representation?
5. What are the properties of linear transformations?
6. How do you determine if a set of vectors forms a basis?
7. How can you represent a vector in terms of different bases?
8. What is the difference between standard basis and an arbitrary basis in \(\mathbb{R}^2\)?

**Tip:** When converting coordinates between bases, always ensure the transformation matrix and vectors are correctly aligned and calculated.

Explore how to apply a linear transformation using the transition matrix [A] from basis B1 to B2. Understand the method to find T(5, 5) in R^2, given the transition matrix [A]_{B1}^{B2} = [[4, 3], [2, -4]].

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