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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.

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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