# Vector Subspaces, don't understand

• clickcaptain
In summary, the problem is to determine whether a given subset of a vector space is a subspace by checking if it satisfies the two properties of closure under addition and closure under scalar multiplication.
clickcaptain
Vector Subspaces, don't understand...

## Homework Statement

Which of the given subsets of the vector space, M23, of all 2 X 3 matrices are subspaces.

(a) [a b c, d 0 0] where b = a + c

## Homework Equations

Theorem 4.3

Let V be a vector space with operations + and * and let W be a nonempty subset of V. Then W is a subspace of V if and only if the following conditions hold

(a) u and v are any vectors in W, then u + v is in W.
(b) If c is any real number and u is any vector in W, then c * u is in W.

## The Attempt at a Solution

First of all I'm not exactly sure what the space R3 exactly is and what to look for.

Is it all the positive numbers in x,y and z? I know what two properties to apply when trying to figure out if its a subspace but I still don't know exactly what to look for.

If someone could explain how to look at this problem, anything about vector spaces, or point me in the direction of a good website about them that would be greatly appreciated...i have yet to find one that I like. Thanks!

Your problem has nothing to do with R^3. If you take two such matrices M1 and M2 with entries a1, b1, c1, d1 (with b1=a1+c1) and a2, b2, c2, d2 (with b2=a2+c2) and add M1+M2 getting a third matrix M3 (so e.g. a3=a1+a2, etc), is it still true that b3=a3+c3? If so, that's your property (a).

Yeah, what you're really trying to do is to determine is if the addition of any 2 elements in the W gives an element in W. Similarly in (b), you are trying to see if the '*' of a real number c to an element of W results in an element contained in W as well.

## What is a vector subspace?

A vector subspace is a subset of a vector space that satisfies all of the properties of a vector space. This means that it is closed under vector addition and scalar multiplication, and contains the zero vector.

## What are the characteristics of a vector subspace?

A vector subspace must contain the zero vector, be closed under vector addition and scalar multiplication, and contain all scalar multiples of its vectors. It must also satisfy the distributive and associative properties of vector operations.

## How can I determine if a set is a vector subspace?

To determine if a set is a vector subspace, you can check if it satisfies all of the properties of a vector subspace. This includes checking if it contains the zero vector, is closed under vector addition and scalar multiplication, and contains all scalar multiples of its vectors.

## What is the purpose of vector subspaces in mathematics?

Vector subspaces are important in mathematics because they allow us to study and generalize vector spaces in a more specific and structured way. They also have applications in many areas, such as linear algebra, physics, and computer science.

## Can a vector subspace be empty?

No, a vector subspace cannot be empty. It must contain at least the zero vector to satisfy the properties of a vector subspace. If a set does not contain the zero vector, it cannot be considered a vector subspace.

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