Math Problem Statement

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?

Solution

We are given the following:

  • Vector v=[1,2]v = [1, 2]
  • Matrix A=2[12−121212]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 (A8)v(A^8) v.

Step 1: Simplifying the matrix AA

The matrix AA can be rewritten as:

= \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.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Operations
Rotations
Scaling Transformations

Formulas

Matrix multiplication formula
Rotation matrix
Scaling factor

Theorems

Rotation matrix properties
Powers of matrices in transformations

Suitable Grade Level

Undergraduate Level (Linear Algebra Course)

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