Let v= [1,2] be a vector in the plane and let A = 2[[1/sqrt(2), - 1/sqrt(2)], [1/sqrt(2), 1/sqrt(2)]]. What is (A^8)v?

We are given the following:

- Vector \( v = [1, 2] \)
- Matrix \( A = 2 \begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \)
- We need to find \( (A^8) v \).

### Step 1: Simplifying the matrix \( A \)
The matrix \( A \) can be rewritten as:
\[
A = 2 \begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix}
   = \begin{bmatrix} \sqrt{2} & -\sqrt{2} \\ \sqrt{2} & \sqrt{2} \end{bmatrix}.
\]

### Step 2: Diagonalization of \( A \)
Matrix \( A \) can be related to a rotation and scaling transformation. To proceed with powers of \( A \), we attempt to diagonalize it. 

Since \( A \) involves rotations, one approach is to observe that \( A \) represents a scaling by \( 2 \) and a rotation by 45° (counterclockwise). Powers of such matrices can often be simplified by recognizing this pattern.

### Step 3: Powers of a rotation matrix
The matrix \( A \) corresponds to a rotation by 45° with a scaling factor. Each power of \( A \) rotates the vector by additional multiples of 45° while maintaining the scaling factor of \( 2 \).

Thus, \( A^8 \) represents a rotation by \( 8 \times 45^\circ = 360^\circ \), which brings the vector back to its original orientation. Since \( A \) scales vectors by a factor of \( 2 \), \( A^8 \) scales the vector by \( 2^8 = 256 \).

### Step 4: Applying \( A^8 \) to the vector \( v \)
Since \( A^8 \) is a scaled identity matrix (after a full rotation), we have:
\[
(A^8) v = 256 v = 256 \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 256 \\ 512 \end{bmatrix}.
\]

### Final Answer:
\[
(A^8) v = \begin{bmatrix} 256 \\ 512 \end{bmatrix}.
\]

Would you like any further details or explanations?

Here are 5 related questions for further exploration:
1. What is the geometric interpretation of matrix powers in transformations?
2. How can we diagonalize a 2x2 matrix to simplify power computations?
3. What happens to the vector when applying a rotation matrix multiple times?
4. How do you compute powers of a general 2x2 matrix?
5. What is the significance of scaling in linear transformations?

**Tip:** When dealing with powers of a matrix, always check if the matrix represents a rotation or scaling, as it can significantly simplify the computation.

Explore how to compute (A^8)v where A is a rotation and scaling matrix. This solution breaks down matrix multiplication, diagonalization, and understanding matrix powers in transformation. Ideal for undergraduate students studying linear algebra.

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