Multiplication by a matrix in GL rotates a plane's basis?

In summary, the conversation discusses the relationship between a k-dimensional plane defined by a matrix A and a matrix g that rotates the basis within the plane. It is stated that gA defines the same plane as A because g simply rotates the basis within the plane. This is proven by showing that a vector normal to the plane is also normal to the plane defined by gA, and since this holds for all basis vectors of the null space of A, the planes must be the same.
  • #1
joej24
78
0
Let [itex] A = (a_{ij}) [/itex] be a [itex] k\times n[/itex] matrix of rank [itex] k [/itex].
The [itex] k [/itex] row vectors, [itex] a_i [/itex] are linearly independent and span a [itex]k[/itex]-dimensional plane in [itex] \mathbb{R}^n [/itex].

In "Geometry, Topology, and Physics" (Ex 5.5 about the Grassmann manifold), the author states that for a matrix [itex] g\in \textrm{GL}(k,\mathbb{R}) [/itex],
[itex] \overline{A} = gA [/itex] defines the same plane as [itex] A [/itex] because [itex] g [/itex] simply rotates the basis within the [itex] k [/itex]-plane.

I'm having trouble seeing this.
 
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  • #2
Let ##r_j## denote the ##j##th row of ##A##, and consider a vector ##v## that is normal to the ##k##-dimensional plane. Note that ##v## must be perpendicular to all ##r_j##, so ##r_j\cdot n=0##.

Then ##Av=0## since the ##j##th component of ##An## is ##r_j\cdot v##.

So ##\bar Av=(gA)v=g(Av)=g\mathbf 0=\mathbf0##. So ##v## is also normal to the plane defined by ##\bar A##. Since that holds for all ##(n-k)## basis vectors of the null space of ##A##, and the rank of ##\bar A## is the same as that of ##A##, the plane (its rowspace) must be the same.
 
  • #3
Thank you, I understand now. When you say
andrewkirk said:
Since that holds for all ##(n-k)## basis vectors of the null space of ##A##
this means that all the ## v ## perpendicular to the ## k##-dimensional plane satisfy ## \overline{A} v = 0##.
 

1. What is GL and how does it relate to matrix multiplication?

GL stands for General Linear group and it consists of all invertible matrices. In the context of linear algebra, GL is used to represent all possible transformations that preserve the structure of a vector space. Matrix multiplication in GL allows us to perform these transformations on a given set of vectors.

2. How does matrix multiplication in GL rotate a plane's basis?

When a matrix in GL is multiplied with a vector, it results in a new vector that is a linear combination of the original vector and other basis vectors. This new vector represents the transformed version of the original vector. Repeating this process with multiple vectors in a plane results in a rotated basis for the plane.

3. Can any matrix in GL rotate a plane's basis?

No, not all matrices in GL can rotate a plane's basis. The determinant of the matrix must be non-zero in order to be able to perform a rotation. This ensures that the transformation is invertible and preserves the structure of the vector space.

4. What are the properties of a matrix in GL that allow it to rotate a plane's basis?

In addition to having a non-zero determinant, a matrix in GL must also be square and have real entries in order to rotate a plane's basis. The matrix must also have an inverse, which allows for the transformation to be undone and the original basis to be restored.

5. How is matrix multiplication in GL used in real-world applications?

Matrix multiplication in GL is widely used in computer graphics, robotics, and physics. It allows for efficient transformation of coordinates and vectors in 3D space. It is also used in data analysis and machine learning, where matrices in GL are used to represent and manipulate large datasets.

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