What values of a and b make S a subspace of R4?

  • MHB
  • Thread starter Johnathon1
  • Start date
  • Tags
    Subspace
In summary, S is a subspace of R4 when a = 1 and b = 0, as this is the only combination of values that satisfies all three equations and makes S closed under addition and scalar multiplication.
  • #1
Johnathon1
2
0
S is the set of solutions for the set of three equations...

x + (1 - a)y-1 + 2z + b2w = 0
ax + y - 3z + (a - a2)|w| = a3 - a
x + (a - b)y + z + 2a2w = b

I worked out...

The first equation is a subset of R4 when a = 1, b is any real.

The second equation is a subset of R4 when a = 1 or a = 0.

The third equation is a subset of R4 when b = 0 and a is any real.

Now, I'm trying to work out the values of a and b that make S a subspace of R4.

Isn't S only a subspace for the values of a and b that are common to all three equations?
 
Physics news on Phys.org
  • #2
So in this case, S would only be a subspace when a = 1 and b = 0. This is because for a set to be a subspace, it must be closed under addition and scalar multiplication, meaning that if x and y are in the set, then ax + by must also be in the set.

In this case, if a = 1 and b = 0, all three equations will have the same coefficients and will be satisfied by the same set of solutions, making S a subspace of R4. However, if a and b have different values, the equations will have different coefficients and will not have the same set of solutions, making S not a subspace of R4.

Therefore, S is only a subspace of R4 when a = 1 and b = 0.
 

FAQ: What values of a and b make S a subspace of R4?

What is a subspace of three equations?

A subspace of three equations is a set of three linear equations that can be solved simultaneously to find a common solution. This solution represents a point in three-dimensional space that satisfies all three equations.

What are the properties of a subspace of three equations?

A subspace of three equations must have three independent equations, meaning that none of the equations can be derived from the others. Additionally, the equations must be linear and have the same number of variables.

How is a subspace of three equations represented geometrically?

A subspace of three equations can be represented geometrically as a point, line, or plane in three-dimensional space. The equations define the coordinates of this point, line, or plane.

What is the purpose of finding a subspace of three equations?

Finding a subspace of three equations allows us to solve for a common solution that satisfies all three equations. This can be useful in various scientific and mathematical applications, such as finding the intersection of three planes or determining the equilibrium point of a system of three variables.

How is a subspace of three equations related to linear algebra?

Subspaces of three equations are a fundamental concept in linear algebra. They represent a set of equations that can be solved using matrix operations and can be used to represent points, lines, and planes in three-dimensional space. Additionally, the properties of subspaces of three equations are closely related to the properties of vector spaces in linear algebra.

Similar threads

MHB What Values of m and n Make These Sets a Subspace of R^4?
Replies
1
Views
1K
MHB Decide if the sets are subspaces or affine subspaces
Replies
3
Views
2K
I 2 and 3 dimensional invariant subspaces of R4
Replies
1
Views
2K
MHB Subset that satisfies all but one axioms of subspaces
Replies
16
Views
2K
MHB Subsets of $\mathbb{R}^2$ Satisfying S2 and S3 but Not S1: Empty Set
Replies
6
Views
2K
MHB Can These Mathematical Sets Be Considered Subspaces?
Replies
1
Views
1K
I Subspaces in Axler's Famous Textbook
Replies
3
Views
2K
MHB Intersection between line and plane in R4-RN
Replies
17
Views
3K
I Understanding Subspaces: Definition and Examples | PF Study Guide
Replies
3
Views
2K
MHB Dimension & Subspace of $\mathbb{R}^3$
Replies
8
Views
2K

Hot Threads

Recent Insights

Back
Top