The general purpose of algebra is to find solutions to systems of equations, which can be modeled in various forms. These equations have applications in finance, neural networks, and science. One thing remains important across all applications: there are unknown variables represented in the algebra..
Equation forms can be complex, but the most common type in real life is the linear equation. Consequently, all equations can be represented in matrix form as shown below:
This scenario involves multiple unknown variables and an expected output. Using matrix A, we can transform these unknown variables to achieve the desired result.
The way we think about solutions to this equation depends on whether the transformation associated with squishes all of space into a lower dimension, like a line or a point, or if it leaves everything spanning the full two dimensions where it started.
The lesson from the previous chapter highlighted two cases for matrix ⁍: it can have either a zero determinant or a non-zero determinant.
Higher Dimensions
Irreversibility and Determinants
Explanation of Key Concepts
- Order of a matrix: The order of a matrix represents the total number of elements within it.
- Dimension of a matrix: The dimension of a matrix specifies the number of rows and columns it contains, written as
m×n - Rank of a matrix: The rank of a matrix is the maximum number of linearly independent columns (or rows) in the matrix. It represents the number of columns (or rows) that provide unique information.
- Nullity of matrix: The nullity of a matrix is the number of columns that map to the zero vector (or, informally, the number of "redundant" columns).
- Column Space: The column space (or range) of a matrix is the set of all possible linear combinations of its columns. It represents the span of all columns in the matrix.
- Null Space:
The set of all vectors that, when multiplied by the matrix, result in the zero vector. It represents solutions to the equation
Overview:
In linear algebra, matrices help us understand how space is transformed. This note covers key ideas like column space and rank—which explain the range of outputs possible from a matrix transformation. Additionally, we explore when solutions to equations exist based on these concepts.
1. Column Space
The column space of a matrix is all the outputs the matrix can produce. It's the "reach" of the transformation that the matrix represents. The column space can be:
- A line: Transformation output is "squished" into a single line.
- A plane: Transformation output fills a plane (2D space).
- 3D space: Transformation reaches all 3D space.
Each column in the matrix shows where basic directions in space end up after transformation. The column space is made up of all the directions those columns span, showing what outputs are possible.
2. Understanding Rank
Rank is the number of dimensions in the column space. It tells us how "big" the output space is after transformation.
- Rank 1: Output is squished to a single line (1D).
- Rank 2: Output is squished to a plane (2D).
- Rank 3: Output fills all of 3D space.
A matrix with the highest possible rank (equal to its number of columns) is called full rank, meaning it can cover the full space without squishing it.
3. Solving Equations When Rank is Less Than Full
Even if a matrix’s determinant is zero (and doesn’t have an inverse), a solution may still exist:
- Rank 1 (1D): The output is a line. For a solution, any desired result must lie on this line.
- Rank 2 (2D): The output is a plane. For a solution, the result must be in this plane.
If the desired result doesn’t lie in the column space, then there’s no solution.
4. Examples of Matrix Rank with Diagrams
Example 1: Rank 1 (Transformation to a Line in 2D)
Consider the matrix:
