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

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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.
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Johnathon1
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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?
 
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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.

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