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Vector Spaces Continued

The Dimension of a Vector Space

Definition

The dimension of a vector space is the number of vectors in its basis, which is a linearly independent set that spans the entire space.

Formula

If \( V \) is a vector space with a basis \( \{v_1, v_2, \ldots, v_n\} \), then:

\[ \dim(V) = n \]

Examples

  1. The dimension of \( \mathbb{R}^3 \) is 3.
  2. For the space of polynomials of degree \( \leq 2 \), the dimension is 3 (basis: \( \{1, x, x^2\} \)).

Tips and Tricks

  1. To find the dimension of a vector space, determine the maximum number of linearly independent vectors.
  2. Use the row-reduction method to identify a basis from a set of vectors.
  3. The number of pivot columns in the row echelon form equals the dimension of the column space.

Rank

Definition

The rank of a matrix is the dimension of its column space (or row space), representing the maximum number of linearly independent columns (or rows).

Formula

\[ \text{rank}(A) = \dim(\text{Col}(A)) = \dim(\text{Row}(A)) \]

Examples

  1. For \( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \):

    Row-reduce to find the rank is 2.

  2. For \( B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \):

    Row-reduce to find the rank is 3.

Tips and Tricks

  1. A square matrix is invertible if and only if \( \text{rank}(A) = n \), where \( n \) is the size of the matrix.
  2. Use the singular value decomposition (SVD) to numerically compute the rank for large matrices.

Change of Basis

Definition

Changing the basis of a vector space means expressing vectors or transformations in terms of a different basis.

Transformation Formula

If \( B \) and \( B' \) are bases of \( V \), the change of basis matrix \( P \) satisfies:

\[ [P]_B^{B'} v_B = v_{B'} \]

Steps to Change Basis

  1. Construct the Change of Basis Matrix:

  2. Write the vectors of the new basis \( B' \) as columns in terms of the old basis \( B \).

  3. Transform Coordinates:

  4. Multiply the change of basis matrix with the coordinate vector in the old basis to get the vector in the new basis.

Example

Transform coordinates from the standard basis \( B_s = \{(1, 0), (0, 1)\} \) to \( B = \{(1, 1), (1, -1)\} \).

Solution

  1. Write \( B \) in terms of \( B_s \): \( P = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \)

  2. For a vector \( v = \begin{bmatrix} 3 \\ 1 \end{bmatrix} \),

    compute:\(P^{-1} v = \begin{bmatrix} 0.5 & 0.5 \\ 0.5 & -0.5 \end{bmatrix} \begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 2 \end{bmatrix}\)

Applications of Difference Equations

Definition

Difference equations describe the relationship between consecutive terms in a sequence, often representing discrete systems.

Example

\( x_{n+1} = 3x_n - 4x_{n-1} \).

Steps to Solve

  1. Find the characteristic equation: \( r^2 - 3r + 4 = 0 \).
  2. Solve for roots. If roots are real and distinct, the solution is:
\[ x_n = C_1 r_1^n + C_2 r_2^n \]

Applications

  1. Population Modeling: Predict population growth or decline over discrete time intervals.
  2. Financial Calculations: Compute compound interest, loan payments, etc.
  3. Signal Processing: Analyze discrete signals in time-series data.

Example Problem

Given \( x_{n+1} = 2x_n - x_{n-1} \), solve for \( x_n \).

Solution

  1. Characteristic equation: \( r^2 - 2r + 1 = 0 \).
  2. Roots: \( r = 1 \) (repeated root).
  3. General solution:
\[ x_n = C_1 + C_2 n \]

Applications of Markov Chains

Definition

A Markov chain is a stochastic process with memoryless transitions between states, meaning the probability of transitioning to the next state depends only on the current state.

Transition Matrix

The probabilities of moving from one state to another are represented in a matrix \( P \):

\[ P = \begin{bmatrix} p_{11} & p_{12} & \cdots & p_{1n} \\ p_{21} & p_{22} & \cdots & p_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ p_{n1} & p_{n2} & \cdots & p_{nn} \end{bmatrix} \]

where \( p_{ij} \) is the probability of transitioning from state \( i \) to state \( j \).

Applications

  1. Google PageRank: Determines the importance of webpages based on link structure.
  2. Weather Prediction: Models probabilities of weather transitions (e.g., sunny to rainy).
  3. Queueing Systems: Analyzes customer arrival and service processes.

Example

A Markov chain with states \( A \) and \( B \):

\[ P = \begin{bmatrix} 0.7 & 0.3 \\ 0.4 & 0.6 \end{bmatrix} \]

Problem

Find the steady-state distribution.

Solution

  1. Solve \( \pi P = \pi \):
\[ \begin{bmatrix} \pi_A & \pi_B \end{bmatrix} \begin{bmatrix} 0.7 & 0.3 \\ 0.4 & 0.6 \end{bmatrix} = \begin{bmatrix} \pi_A & \pi_B \end{bmatrix} \]
  1. Solve the system of equations:
\[ 0.7\pi_A + 0.4\pi_B = \pi_A \\ 0.3\pi_A + 0.6\pi_B = \pi_B \\ \pi_A + \pi_B = 1 \]
  1. Solution: \( \pi_A = 0.571, \pi_B = 0.429 \).