Understanding Matrix Geometry: Exploring Null Rows in a 4x5 Matrix

[DespicableMe]

Homework Statement

Given a 4x5 matrix, what could a row of 0's represent geometrically?

The Attempt at a Solution

Given a scalar PLANE equation, you could make a 3x3 matrix and solve the system of equations.

A row of zeroes there could represent a consisten but dependent solution,
The planes would intersect in a line.
But PLANES only have 3 variables.

we were given FOUR equations and FIVE variables and we had to make a matrix out of that.

What could a row of 0's in a 4x5 matrix represent?

 

[HallsofIvy]

A 4 by 5 matrix (4 columns, 5 rows) maps a 4 dimensional space to a subspace (of dimension of at most 4) of a 5 dimensional space. In particular, it represents a linear transformation from 4 dimensional space to 5 dimensional space with specific given orthonormal bases for each space. If one row consists entirely of 0s, that basis vector (of the 4 dimensional space) plays no part in the definition of the linear transformation. That means that it could be thought of as representing a linear transformation from the three dimensional subspace orthogonal to that particular basis. In particular, its range is an at most three dimensional subspace of the 5 dimensional space.

 

[Mark44]

Homework Statement

Given a 4x5 matrix, what could a row of 0's represent geometrically?

The Attempt at a Solution

Given a scalar PLANE equation, you could make a 3x3 matrix and solve the system of equations.

A row of zeroes there could represent a consisten but dependent solution,
The planes would intersect in a line.
But PLANES only have 3 variables.

Planes in three-dimensional space can have one, two, or three variables. For example, the equation z = 1 represents a plane in 3D. So also does the equation 2x - y = 0.

If you have three linear equations in three variables, each equation represents a plane in 3D space. You can solve this system by setting up a 3 x 4 augmented matrix and row-reducing it to find the solution(s).

In higher dimensions a linear equation represents something called a hyperplane. For example, 2x + 3y + z -5w = 3 is a hyperplane in four dimensions. A system of two equations like this in four variables represents two hyperplanes that could a) not intersect at all (no solution) b) intersect in a line (infinite number of solutions, all of which lie on a line), c) intersect at every point on either hyperplane (infinite number of solutions again).

In matrix form a system of two equations in four variables could be represented by a 2 x 5 augmented matrix.

Now for your question, if you have a 4 x 5 matrix with a row of zeroes, a question I would ask is whether this is an augmented matrix. If so, we're in four dimensions. If not, we're in five dimensions.

we were given FOUR equations and FIVE variables and we had to make a matrix out of that.

What could a row of 0's in a 4x5 matrix represent?

The fact that there are five columns doesn't necessarily imply there are five variables. If the matrix is a 4 x 5 augmented matrix, there are four variables.

If the matrix is not an augmented matrix, then there are five variables, and the assumption would be that the constants on the right sides of all four equations are all zero. So if there are three nonzero rows, the geometry is that you have three hyperplanes in 5D space. A row of zeroes represents the equation 0x + 0y + 0z + 0w + 0v = 0. Since this equation is true for any 5-tuple of numbers (i.e., any point in 5D space) it doesn't place any restrictions on the solution set.