# Vector subspace as the space of solutions to matrix multiplication

• uman
In summary, a vector subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. A vector space is a mathematical structure consisting of vectors and operations of vector addition and scalar multiplication. Matrix multiplication is an operation that takes two matrices and produces a new matrix, often used to represent linear transformations and solve systems of linear equations. A vector subspace can be thought of as the space of all possible solutions to a system of linear equations represented by matrix multiplication, such as the null space, column space, row space, and left null space of a matrix.
uman
Given a subset W of a vector space V = F^n (for some field F), prove that W is the subspace of solutions of the matrix equation AX = 0 for some A.

Why?

Seriously, this sounds like a homework problem so it should be in the homework section.

In any case, define the orthogonal complement of W and let A be the projection onto that complement.

I would just solve for A myself.

## What is a vector subspace?

A vector subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. This means that if you take any two vectors from the subspace and add them together, the result will also be in the subspace. Similarly, if you multiply any vector in the subspace by a scalar (a real number), the result will also be in the subspace.

## What is a vector space?

A vector space is a mathematical structure that consists of a set of vectors and operations of vector addition and scalar multiplication. It is often used in linear algebra to represent physical quantities and mathematical objects. Examples of vector spaces include Euclidean space, function spaces, and solution spaces of linear equations.

## What is matrix multiplication?

Matrix multiplication is an operation that takes two matrices and produces a new matrix. It is defined as the dot product of the rows of the first matrix and the columns of the second matrix. Matrix multiplication is used to represent linear transformations and to solve systems of linear equations.

## How is a vector subspace related to matrix multiplication?

A vector subspace can be thought of as the space of all possible solutions to a system of linear equations represented by matrix multiplication. This is because the equations in the system can be written in the form of a matrix equation, and the solutions to that equation will form a vector subspace. This is useful in solving systems of equations and understanding the properties of linear transformations.

## What are some examples of vector subspaces as spaces of solutions to matrix multiplication?

Some examples include the null space and column space of a matrix, which are subspaces that represent the solutions to homogeneous and non-homogeneous systems of linear equations, respectively. Additionally, the row space and left null space of a matrix also represent subspaces of solutions to matrix multiplication.

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