What Values of m and n Make These Sets a Subspace of R^4?

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In summary: Contains the zero vector: Plugging in all zeros for x, y, z, and w, we get 0 + (1 - m)(0)-1 + 2(0) + n2(0) = 0. Therefore, the set contains the zero vector.Since the set satisfies all three properties, it is a subspace of R (four dimensions). This means there exist values for m and n that make the set a subspace.2. For the second set, we have mx + y - 3z + (m - m2)|w| = m3 - m. Similarly, we need to check if it satisfies the three properties of subspaces.- Closure under addition: Let's
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Johnathon1
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D is the set and the set contains the solutions to

x + (1 - m)y-1 + 2z + n2w = 0

I'm trying to find m, n values which means the set is a subspace of R (four dimensions).

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Similarly, trying to find the m, n values that makes the following two expressions two separate subspaces, too.

mx + y - 3z + (m - m2)|w| = m3 - m

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x + (m - n)y + z + 2m2w = n

I've been reviewing the three properties of subspaces over and over and don't know how to apply them in these scenarios.
 
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Hi there,

I am happy to help you understand the concept of subspaces and how they apply to the given scenarios.

First, let's review the three properties of subspaces:

1. Closure under addition: This means that if two vectors are in the subspace, their sum must also be in the subspace.

2. Closure under scalar multiplication: This means that if a vector is in the subspace, multiplying it by any scalar will result in a vector that is still in the subspace.

3. Contains the zero vector: This means that the subspace must contain the zero vector (a vector where all components are equal to zero).

Now, let's apply these properties to the given sets:

1. For the first set, we have x + (1 - m)y-1 + 2z + n2w = 0. To determine if this set is a subspace, we need to see if it satisfies the three properties mentioned above.

- Closure under addition: Let's take two vectors from the set, say (x1, y1, z1, w1) and (x2, y2, z2, w2). If we add them together, we get (x1 + x2, y1 + y2, z1 + z2, w1 + w2). Plugging this into the original equation, we get:

(x1 + x2) + (1 - m)(y1 + y2)-1 + 2(z1 + z2) + n2(w1 + w2) = 0

Since the original equation is true for both (x1, y1, z1, w1) and (x2, y2, z2, w2), it must also be true for their sum. Therefore, the set is closed under addition.

- Closure under scalar multiplication: Let's take a vector from the set, say (x, y, z, w). If we multiply it by a scalar, say k, we get (kx, ky, kz, kw). Plugging this into the original equation, we get:

kx + (1 - m)(ky)-1 + 2kz + n2(kw) = 0

Since the original equation is true for (x, y, z, w), it must also be true for its scalar multiple. Therefore, the set is closed under scalar multiplication
 

FAQ: What Values of m and n Make These Sets a Subspace of R^4?

What are the three subspace problems?

The three subspace problems refer to three fundamental questions in linear algebra: the column space problem, the null space problem, and the row space problem. These problems involve finding the basis and dimension of each subspace for a given matrix.

Why are the three subspace problems important?

Understanding the three subspace problems is crucial in many areas of science and engineering, especially in the fields of data analysis and machine learning. These problems help us understand the structure and properties of a matrix, and can be used to solve a wide range of mathematical and practical problems.

How do you solve the three subspace problems?

To solve the three subspace problems, you need to use techniques such as Gaussian elimination, matrix operations, and linear transformations. You can also use the concept of linear independence and orthogonality to find the basis of each subspace.

Can the three subspace problems have multiple solutions?

Yes, the three subspace problems can have multiple solutions. This is because there can be more than one basis for a subspace, and different methods of solving the problems may yield different solutions. However, the dimension of each subspace will always be the same.

How are the three subspace problems related to each other?

The three subspace problems are closely related to each other, as they all involve finding different subspaces of a matrix. The column space problem and the row space problem are dual to each other, while the null space problem is orthogonal to both. Together, these problems provide a comprehensive understanding of the structure of a matrix.

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