Linear subspace

In mathematics, especially in linear algebra, a linear subspace (also called a vector subspace) is a special kind of space that sits inside a larger space. Think of it like a smaller, perfectly flat surface inside a bigger, 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 closed under addition and scalar multiplication, meaning if you take any two vectors in the subspace and add them, or multiply one by a number, the result stays in the subspace. This makes it behave nicely and keeps its shape.

Subspaces are important because they help us break down complicated problems into simpler parts. They are used in many areas, from computer graphics to 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 these 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. Imagine you have a big box of building blocks, and a smaller box inside it that still follows all the same rules for stacking and combining blocks. This smaller box is like a linear subspace.

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, and 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 kind of vector space that sits inside a larger vector space. Think of it like a flat surface inside a bigger space — it always passes through the origin (the zero point).

You can describe a subspace in several ways. One way is by solving equations where all the solutions 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, the set of all (x, y, z) that satisfy both equations forms a one-dimensional subspace. Another way is by using "spans" — combining vectors using multipliers. If you multiply vectors by numbers and add them together, all the results form a subspace too. For instance, combining 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 contained within another, the larger one must have equal or greater dimension. 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

Most ways to work with subspaces use something called row reduction. This is a method for changing a matrix using special steps, called elementary row operations, 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 key task is finding a "basis" for the row space of a matrix. This means finding certain rows that can be used to describe all the rows in the matrix. We do this by putting the matrix into a simpler form and then 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 extract the needed information from the simpler form.