Linear Algebra: Prove that the set of invertible matrices is a Subspace

In summary, the question is whether U = {A| A \in nℝn, A is invertible} is a subspace of nℝn, the space of all nxn matrices. The attempt at a solution involves proving this using the regular operations of vector addition and scalar multiplication. However, the set is not closed under these operations as shown by the examples of 0*I and I + (-I). It is unclear if there is a way to prove this without assuming the usual vector operations.
  • #1
Millacol88
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Homework Statement


Is U = {A| A [itex]\in[/itex] nℝn, A is invertible} a subspace of nℝn, the space of all nxn matrices?

The Attempt at a Solution


This is easy to prove if you assume the regular operations of vector addition and scalar multiplication. Then the Identity matrix is in the set but 0*I and I + (-I) are not, so its not closed under vector addition or scalar multiplication. Is there a way to prove this without assuming the usual vector operations?
 
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  • #2
Millacol88 said:

Homework Statement


Is U = {A| A [itex]\in[/itex] nℝn, A is invertible} a subspace of nℝn, the space of all nxn matrices?

The Attempt at a Solution


This is easy to prove if you assume the regular operations of vector addition and scalar multiplication. Then the Identity matrix is in the set but 0*I and I + (-I) are not, so its not closed under vector addition or scalar multiplication. Is there a way to prove this without assuming the usual vector operations?

Not until they tell you what the alternative addition and multiplication operations are. I don't think you have to worry about that. Your examples are just fine.
 

FAQ: Linear Algebra: Prove that the set of invertible matrices is a Subspace

1. What is a subspace?

A subspace is a subset of a vector space that is closed under addition and scalar multiplication. This means that when two vectors in the subspace are added together or multiplied by a scalar, the result is still in the subspace.

2. How do you prove that a set is a subspace?

To prove that a set is a subspace, you must show that it satisfies the three properties of closure under addition, closure under scalar multiplication, and contains the zero vector. For the set of invertible matrices, you must also prove that the inverse of each matrix in the set is also in the set.

3. What is an invertible matrix?

An invertible matrix is a square matrix that has an inverse matrix. This means that when the matrix is multiplied by its inverse, the result is the identity matrix. In other words, the inverse "undoes" the original matrix.

4. How can you prove that a matrix is invertible?

A matrix is invertible if its determinant is not equal to zero. This means that the matrix has a unique solution and can be "undone" by its inverse. You can also use other methods such as row reduction or calculating the rank of the matrix to prove invertibility.

5. Why is the set of invertible matrices considered a subspace?

The set of invertible matrices satisfies the three properties of a subspace: closure under addition, closure under scalar multiplication, and contains the zero vector. Additionally, the inverse of each matrix in the set is also in the set, making it closed under inverses. This makes it a subset of the vector space of all square matrices, and therefore a subspace.

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