Linear subspace

In mathematics, especially in linear algebra, a linear subspace (also called a vector subspace) is a special kind of space inside a bigger space. Imagine a small, flat surface inside a larger, more complex shape. For example, a straight line or a flat plane can be a subspace inside three-dimensional space.

A linear subspace must follow certain rules. It has to stay the same when you add any two vectors in the subspace or when you multiply one by a number. This helps it keep its shape.

Subspaces are important because they help us solve hard problems by breaking them into simpler parts. They are used in many areas, like computer graphics, physics, and engineering. By studying these smaller spaces, scientists and engineers can understand bigger systems better.

In everyday language, people often just call them "subspaces," but the word "linear" reminds us that they follow special straight-line rules. This idea helps connect many different parts of math and science in a clear and useful way.

Definition

A linear subspace is a special part of a bigger mathematical space called a vector space. Think of it like a small box inside a big box of building blocks. This smaller box still follows all the same rules for stacking and combining blocks.

The smallest subspaces are very simple: one contains just the “zero” element, and the other is the entire space itself. These simple subspaces are called trivial subspaces.

Examples

A linear subspace is a smaller space inside a bigger space that follows special rules. Here are some simple examples:

1. Imagine all points in 3D space where the last number is zero, like (1, 2, 0). If you add any two such points or stretch/shrink one, the result still has a zero in the last spot. So this set is a subspace.
2. In a flat 2D plane, look at points where the x and y numbers are the same, like (3, 3). Adding two of these points or stretching one still gives points where x equals y. This is also a subspace.

These ideas show how smaller spaces can sit inside bigger ones while keeping the same structure. Main articles: real coordinate space, real numbers, Cartesian plane, homogeneous system of linear equations, continuous functions, differentiable functions, functional analysis.

Properties of subspaces

A subspace is a special part of a bigger mathematical space called a vector space. It must follow certain rules: it can’t be empty. If you add any two elements in the subspace or multiply an element by a number, the result should still stay in the subspace. This means the subspace is “closed” under addition and multiplication by numbers.

In more complex spaces, a subspace doesn’t always have to stay closed in a topological sense. However, if the subspace has a limited number of dimensions — known as being finite-dimensional — it will always stay closed. The same rule applies to subspaces defined by a limited number of continuous linear measurements.

Descriptions

A linear subspace is a special type of vector space that fits inside a larger vector space. Imagine it like a flat surface inside a bigger space — it always goes through the origin (the zero point).

You can describe a subspace in a few ways. One way is by solving equations where all the answers add up to zero. For example, if you have equations like x + 3_y_ + 2_z_ = 0 and 2_x_ − 4_y_ + 5_z_ = 0, all the pairs (x, y, z) that satisfy both equations make up a one-dimensional subspace. Another way is by using "spans" — combining vectors with numbers. If you multiply vectors by numbers and add them together, all the results make a subspace. For example, mixing the vectors (2, 5, −1) and (3, −4, 2) in different ways creates a two-dimensional subspace.

Operations and relations on subspaces

The relationships between subspaces can be understood through several key ideas. First, subspaces can be compared using inclusion. If one subspace is inside another, the bigger one must have equal or more dimensions. For example, a line fits inside a plane, but a plane cannot fit inside a line.

Subspaces also combine in interesting ways. Two subspaces can intersect, meaning they share some vectors. This intersection is itself a subspace. Subspaces can also be added together, creating a new subspace that contains all vectors from both original subspaces. For instance, adding two lines in a plane can give the entire plane.

In spaces with angles, like those used in geometry, each subspace has an "orthogonal complement"—another subspace made of vectors at right angles to the original. These complementary subspaces together make up the entire space, and their dimensions always add up to the dimension of the whole space.

Algorithms

We use a method called row reduction to work with subspaces. This method changes a matrix using special steps until it becomes simpler.

Row reduction has some important facts:

1. The changed matrix has the same "null space" as the original.
2. Row reduction does not change the "span" of the row vectors.
3. Row reduction does not change how the column vectors depend on each other.

One important task is finding a "basis" for the row space of a matrix. This means finding certain rows that can describe all the rows in the matrix. We do this by putting the matrix into a simpler form and using the nonzero rows of that simpler form as our basis.

We can also check if a vector belongs to a subspace. We do this by creating a matrix with the basis vectors of the subspace and the vector we are checking, then using row reduction to see if the vector can be made from the basis vectors.

Other tasks include finding a basis for the column space, coordinates for a vector in terms of a basis, and a basis for the null space of a matrix. All these tasks use row reduction to simplify the matrix and then get the needed information from the simpler form.