# Linear Algebra - Find a subspace

• cristina89
So we'll be looking for the span of two vectors to represent W.In summary, we are given a subspace V of the vector space R^{3} defined by two planes. To find a subspace W such that R^{3} is the direct sum of V and W, we need to find a plane perpendicular to V containing the origin. This can be done by solving the system of equations and finding the span of two vectors that represent W.
cristina89

## Homework Statement

V is a subspace of the vector space $R^{3}$ given by:
V = {(x, y, z) E $R^{3}$ / x + 2y + z = 0 and -x + 3y + 2z = 0}
Find a subspace W of $R^{3}$ such that $R^{3}$ = V$\oplus$W

I'm really lost in this. My teacher didn't give any example of how to solve this kind of exercise... Can anyone help me how to start and develope this?

each of those equations defines a plane - what is the intersection of two planes?

lanedance said:
each of those equations defines a plane - what is the intersection of two planes?

A straight line?

Well, if I solve this system:

x = -2y + z
y = -3/5z --> x = -11/5z

Last edited:
To start with, do you know what your looking for ? Do you know what is the direct sum of two subspaces?

So the subspace V is a line through the origin, and can be represented by the span of a single vector parallel to that line.

Now onto Dansure's question...

W will be a plane perpendicular to that line, containing the origin.

## 1. What is a subspace in linear algebra?

A subspace in linear algebra is a subset of a vector space that satisfies the three properties of a vector space: closure under addition, closure under scalar multiplication, and contains the zero vector.

## 2. How do you find a subspace in linear algebra?

To find a subspace in linear algebra, you can use the following steps:

• Identify the vectors or equations that make up the subspace.
• Check if the subspace satisfies the three properties of a vector space.
• If the subspace satisfies the properties, then it is a subspace.

## 3. What is the importance of finding subspaces in linear algebra?

Finding subspaces in linear algebra is important because they allow us to simplify complex vector spaces and focus on specific properties or characteristics of the space. It also helps in solving systems of linear equations and understanding the relationship between different vectors.

## 4. Can a subspace have an infinite number of vectors?

Yes, a subspace can have an infinite number of vectors as long as the three properties of a vector space are satisfied. This is because the three properties allow for the creation of new vectors through addition and scalar multiplication.

## 5. How is a subspace different from a span in linear algebra?

A subspace is a subset of a vector space that satisfies the three properties of a vector space, while a span is the set of all possible linear combinations of a given set of vectors. In other words, a subspace is a type of vector space, while a span is a set of vectors within a vector space.

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