Can vector spaces and their subspaces be visualized effectively?

[Dosmascerveza]

The linear algebra course I'm taking just became very "wordy" and I am having a hard time dealing notions such as subspaces without a diagram. I was thinking Venn diagrams could be used to visualize relationships between subspaces of vector spaces. Has this been a useful way to organize the concepts for you?

 

[jbunniii]

I don't think Venn diagrams are all that useful because they lack the structure implied by a vector space.

In low dimensions, it's probably easiest to visualize in terms of Euclidean space. Any 3-dimensional vector space is isomorphic to either \mathbb{R}^3 or \mathbb{C}^3, depending on your scalar field. The two-dimensional subspaces are precisely the planes that pass through the origin, and the one-dimensional subspaces are precisely the lines that pass through the origin. In higher dimensions the idea is the same but it's hard/impossible to visualize.

For linear maps, I find it easiest to visualize what the map does to a unit "sphere" of appropriate dimension. The singular value decomposition implies that EVERY linear map is the result of a rigid rotation, followed by individual stretch or compress factors along the coordinate axes, followed by another rigid rotation. Thus a sphere is mapped to an ellipsoid, not necessarily aligned with the coordinate axes.

 

[Dosmascerveza]

Interesting. Please excuse me if I seem to be an echo chamber but I just want to make sure that I have an understanding of your strategy... So in R2 all lines through the origin are subspaces and form a subspace. in R3 all planes through the origin would form a subspace and would themselves be a subspaces. And higher dimensional vector spaces, and their subspaces would be impossible to visualize. Venn diagrams are ineffective tools for the general case. As to linear maps, do you mean linear transformations? If so do these linear maps send a vector from one dimension to another? for exampls. Given v==(a1,a2) is a vector in R2 and T:v from R2--> R3 s.t. T{(a1,a2)}=(a1,a2,a3)?

 

[HallsofIvy]

Interesting. Please excuse me if I seem to be an echo chamber but I just want to make sure that I have an understanding of your strategy... So in R2 all lines through the origin are subspaces and form a subspace. in R3 all planes through the origin would form a subspace and would themselves be a subspaces. And higher dimensional vector spaces, and their subspaces would be impossible to visualize. Venn diagrams are ineffective tools for the general case. As to linear maps, do you mean linear transformations? If so do these linear maps send a vector from one dimension to another? for exampls. Given v==(a1,a2) is a vector in R2 and T:v from R2--> R3 s.t. T{(a1,a2)}=(a1,a2,a3)?

They send a vector in one vector space to a vector space. That "range" space might have different dimension or the same dimension. Indeed it might be the same vector space as the original "domain" space.

For example T(a1,a2)= (-a2, a1, a1+a2) maps R² to R³ as in your example. T(a1,a2,a3)= (a1-a2, a2-2a3) maps R³ to R². T(a1, a2)= (a2+a1, 3a1- 2a2) maps R² to R².

Oh and don't forget the "identity" transformation on any vector space that maps each vector onto itself.

 

[Fredrik]

As to linear maps, do you mean linear transformations?

"Linear map", "linear function", "linear transformation", "linear operator"...all these terms mean the same thing. "Transformation" seems to be used a lot in the context of finite-dimensional vector spaces, but "operator" is used more in quantum mechanics and functional analysis. Some people like to use "operator" only when the function is from a vector space V into the same vector space V. If V is a vector space over a field F, and the function we're talking about is from V into F, the preferred term is "linear functional" (even when F also has the structure of a vector space), especially when the members of the vector space are functions. "linear form" is an alternative to "linear functional", but the term "form" is usually only used for multilinear maps (e.g. the inner product on a real vector space).