Unit 5: Linear Algebra

Elementary Transformation of Matrices

These are operations on the rows (or columns) of a matrix that do not change its fundamental properties, like its rank or the solution set of the linear system it represents.

The Three Elementary Row Operations (EROs)

1. Swap: Interchange any two rows. (Rᵢ ↔ Rⱼ)
2. Scale: Multiply a row by a non-zero scalar. (kRᵢ → Rᵢ)
3. Add: Add a multiple of one row to another row. (Rᵢ + kRⱼ → Rᵢ)

An elementary matrix is a matrix obtained by performing a single ERO on an identity matrix.

Echelon and Canonical Forms

Row Echelon Form (REF)

A matrix is in row echelon form if it satisfies:

1. All non-zero rows (rows with at least one non-zero element) are above any all-zero rows.
2. The first non-zero entry in a row (called the pivot or leading entry) is to the right of the pivot in the row above it.
3. All entries in a column below a pivot are zero.

Canonical Form (Row Reduced Echelon Form - RREF)

A matrix is in canonical form (RREF) if it is in REF and also satisfies:

1. Every pivot is 1.
2. The pivot is the only non-zero entry in its column (i.e., all entries above the pivot are also zero).

Every matrix is row-equivalent to one and only one RREF.

Rank of a Matrix

The rank of a matrix A, denoted `rank(A)`, is the number of non-zero rows in its row echelon form.

The rank represents the number of linearly independent rows (or columns) in the matrix. It is a measure of the "non-degeneracy" of the system.

Consistency of Linear Equations (Rouché-Capelli Theorem)

We analyze the system of linear equations Ax = b, where A is the coefficient matrix, x is the variable vector, and b is the constant vector.

We form the augmented matrix `[A | b]`.

Test for Consistency:

1. If `rank(A) < rank([A | b])`:
   The system is inconsistent (NO solution). This happens when the echelon form has a row like `[0 0 0 | k]` where k is non-zero.
2. If `rank(A) = rank([A | b])`:
   The system is consistent (at least one solution).

Types of Consistent Solutions

If the system is consistent, let `n` be the number of variables (columns of A).

• If `rank(A) = n`: The system has a unique solution.
• If `rank(A) < n`: The system has infinitely many solutions. The number of free variables (parameters) is `n - rank(A)`.

Row Reduced Echelon Form (RREF)

As defined in Unit-V, this is the "canonical form". It is a unique form for every matrix, obtained by applying EROs.

Example:

Matrix A: [ 1 2 3 ]
[ 4 5 6 ] RREF: [ 1 0 -1 ]
[ 0 1 2 ]

Finding the Inverse of a Matrix (RREF Method)

To find the inverse `A⁻¹` of a square matrix A, we use the RREF method.

Algorithm

1. Create an augmented matrix `[A | I]`, where `I` is the identity matrix of the same size.
2. Perform elementary row operations on this entire augmented matrix.
3. Your goal is to transform the left side (A) into the identity matrix (I).
4. If you succeed, the right side will have transformed into the inverse, `A⁻¹`. [A | I] ---EROs---> [I | A⁻¹]
5. If you cannot get `I` on the left side (i.e., you get a row of all zeros), then the matrix A is singular (not invertible) and `A⁻¹` does not exist.

Solving Linear Equations (Gaussian Elimination)

This is a systematic method for solving a system `Ax = b`.

Method

1. Write the augmented matrix `[A | b]`.
2. Use Elementary Row Operations to reduce this matrix to Row Echelon Form (REF).
3. This new REF matrix represents a simpler, equivalent system of equations.
4. Solve this new system using back substitution.
   • Solve the last non-zero equation for its variable.
   • Substitute this value into the equation above it, and solve for the next variable.
   • Continue "substituting backwards" up to the first equation.