True or false: If W is an invariant subspace of the matrix A, then it is also an invariant subsp

step 1 Hello and welcome to problem 2 .1 .20. Here we are asked three questions, two or false questions

True or false: If W is an invariant subspace of the matrix...

Key Concepts

Invariant Subspace

An invariant subspace for a linear operator is a subspace that is mapped onto itself under the operator. In other words, if you take any vector from the subspace and apply the operator (or matrix) to it, the resulting vector remains in that same subspace. This concept is fundamental in understanding how operators can be decomposed and analyzed with respect to simpler parts of a space.

Matrix Transpose

The transpose of a matrix is obtained by swapping its rows with its columns. This operation is important in linear algebra because it affects many properties of the matrix, such as its symmetry. When considering invariant subspaces, the transpose of a matrix does not, in general, preserve the invariance properties of the original matrix unless the matrix is symmetric (or satisfies certain other conditions), highlighting the nuanced relationship between a matrix and its transpose.

Transcript

00:01 Hello and welcome to problem 2 .1 .20.

00:04 Here we are asked three questions, two or false questions.

00:09 The first one is if the skew symmetric three by three matrices form a subspace.

00:14 Those are the matrices where the transpose is equal to their negation.

00:19 The second is if the ones that are unsymmetric, so those are the ones where the transpose is not equal to itself.

00:29 The third one is the matrices.

00:30 That have a one -one -one in their null space.

00:33 So let's start off with the original one, whether the matrices that have a transpose equal to their negation.

00:44 So let's say we have two elements in this.

00:51 We can just use x and y are elements of the space and and x and y are 3 by 3 matrices and if we add them together this is a we're trying to show vector addition so x plus y transpose must be equal to in the end the negation of their sum so by properties of transpose this is equal to a transpose plus b transpose next we can change them individually.

01:37 We know that a, whoops, let's do a there, should be, sorry, x transpose plus y transpose.

01:49 And we know from the definition that this is simply equal to negative x plus negative y, which is negative x plus negative y.

02:02 And from here, we can factor out a y, or factor a negative.

02:07 So i have negative x plus y, which is what we needed to show that this is consistent under vector addition.

02:19 Next we show that this is consistent under scalar multiplication.

02:26 So if we have a constant c multiplied by x and it's transposed, we want to show that this is equal to the negation of the matrix x.

02:42 So we can show that we will be able to factor a c because constants can be factored out of all matrices.

02:52 So we'll have c times x transpose and by the definition of this subspace or of this proposed subspace we'll have c times negative x.

03:06 This negative is a constant so we can factor it out.

03:10 We'll have negative cx, which is what we desire for scalar multiplication.

03:19 So yes, this is a valid subspace.

03:25 The next one is the unsymmetric matrices in m.

03:30 And right away, you can sort of see that this is different from the others.

03:36 There's a way where under vector addition, if we add two matrices that don't transpose, they have a transpose that's not their original selves, when you add them together, you can have something where the transpose is equal...

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