# Linear algebra: Finding a linear system with a subspace as solution set

• sphlanx
In summary, a subspace of R^3 is given and we are asked to find a homogeneous linear system that has this subspace as its solution set. The subspace is 2 dimensional, so the solution set must have 2 parameters. To find a system with 3 variables and 2 equations, we need a set of equations that satisfy the four conditions: 2a+ 6b+ 2c= P, 2d+ 6e+ 2f= Q, 6a+ 2b+ 2c= P, and 6d+ 2e+ 2f= Q. This can be achieved by solving for four variables in terms of the other four and choosing
sphlanx

## Homework Statement

We are given a subspace of R^3 that is produced by the elements: (2,6,2) abd (6,2,2). We are asked to find (if any) a homogeneous linear system that has this subspace as solution set.

## The Attempt at a Solution

1)The subspace is 2 dimensional so the solution set must have 2 parameters. Also, given the elements that produce the subspace, i guess we want a system with 3 variables and 2 equations.

No clue after that :S

sphlanx said:

## Homework Statement

We are given a subspace of R^3 that is produced by the elements: (2,6,2) abd (6,2,2). We are asked to find (if any) a homogeneous linear system that has this subspace as solution set.

## The Attempt at a Solution

1)The subspace is 2 dimensional so the solution set must have 2 parameters. Also, given the elements that produce the subspace, i guess we want a system with 3 variables and 2 equations.

No clue after that :S
Yes, that's right. You want to equations, say ax+ by+ cz= P and dx+ ey+ fz= Q that are both satisfied by (2,6,2) and (6,2,2). That is, you must have the four equations 2a+ 6b+ 2c= P, 2d+ 6e+ 2f= Q, 6a+ 2b+ 2c= P, and 6d+ 2e+ 2f= Q. That gives you four equations to solve for 8 numbers, but, of course there are many sets of equations that will satisfy this problem. Solve for four of the variables in terms of the other four, then choose whatever numbers you please for those four.

## 1. What is a linear system?

A linear system is a set of equations that can be represented using linear equations, where the variables are raised to the first power. The goal of solving a linear system is to find values for the variables that make all the equations true simultaneously.

## 2. What is a subspace?

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

## 3. How do you find a linear system with a subspace as a solution set?

To find a linear system with a subspace as a solution set, you must first identify the basis vectors of the subspace. Then, you can construct a system of equations using these basis vectors as the columns of the coefficient matrix. The solution set of this system will be the given subspace.

## 4. Can a subspace be the solution set of more than one linear system?

Yes, a subspace can be the solution set of infinitely many linear systems. This is because a subspace can be spanned by different sets of basis vectors, and different combinations of these basis vectors can be used to construct different linear systems with the same solution set.

## 5. How is linear algebra used in real life?

Linear algebra has many applications in fields such as engineering, computer science, economics, and physics. Some real-life examples include using linear transformations to compress images, using matrices to represent and solve systems of equations in economics, and using eigenvectors and eigenvalues to analyze the stability of physical systems.

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