True or false: If V and W are invariant subspaces for the matrix A, then so is (a) V+W (b) V ∩W;

step 1 Hello and welcome to problem 2 .1 .28. Here we're given four true or false questions and we're a

True or false: If V and W are invariant subspaces for the...

Key Concepts

Set Difference of Subspaces

The set difference, which consists of elements in one subspace that are not present in a second subspace, is typically not a subspace. This is because it might not satisfy the necessary conditions for subspaces, such as being closed under addition or scalar multiplication. Consequently, even if the original subspace is invariant under a transformation, its set difference does not automatically inherit invariant subspace properties.

Invariant Subspaces

An invariant subspace for a linear transformation (or matrix) is a subset of the vector space that remains stable under the transformation. This means that when any vector in the subspace is transformed, the result still lies within the same subspace. This concept is fundamental in understanding the structure and behavior of linear operators.

Sum of Subspaces

The sum of two subspaces is the set of all vectors that can be expressed as the sum of one vector from each subspace. When both subspaces are invariant under a linear transformation, the sum is also invariant; this occurs because the action of the transformation on each component vector preserves their membership within their respective invariant subspaces, thus keeping the sum within the combined space.

Intersection of Subspaces

The intersection of two subspaces is the set of vectors that are common to both subspaces. If each subspace is invariant under a linear transformation, then every vector in their intersection remains in both subspaces after the transformation, ensuring that the intersection is also invariant. This is a key property when analyzing the shared structure within invariant subspaces.

Union of Subspaces

The union of two subspaces is the set containing vectors from either subspace. In general, the union of two subspaces does not constitute a subspace unless one of the subspaces is wholly contained within the other, as it may not be closed under addition or scalar multiplication. Therefore, even if the individual subspaces are invariant under a transformation, their union may fail to be an invariant subspace.

Transcript

00:01 Hello and welcome to problem 2 .1 .28.

00:08 Here we're given four true or false questions and we're asked to answer them and give a brief explanation.

00:15 The first one, part a, is are the vectors b, they're not in the column space of a? do this form a subspace? the answer to this is no.

00:28 The reason is because if the column space forms a subspace, which we know, then it will contain the zero vector.

00:37 So c of a contains a zero.

00:45 And we know all subspaces have to contain a zero.

00:49 So if those vectors that are not in the column space of a, they don't contain a zero, then that means that it's not a subspace.

01:01 So not c of a is not a subspace.

01:13 All right, let's go to part b.

01:15 If the column space of a contains only the zero vector, then a is the zero matrix.

01:23 The column space of a matrix is just all the linear combinations of that matrix.

01:30 So if the column space is only the zero vector, then that means a only spans the zero vector, which is true only if a is the zero matrix.

01:40 So b is true.

01:44 See, a only spans the zero.

01:51 About c.

01:54 So the column space of 2a is equal to the column space of a.

01:59 This is also true.

02:02 The way we can see this is, let's say we have a, x equals b.

02:08 So that means that for the given matrix a, there exists an x such that we have the solution b, the solution vector b...

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