Quick question on span (linear algebra)

[dark marsh]

If the span of two vectors is a line (they are mulitples of each other) and B is the linear transformation of those two vectors, is B still in the span of those two vectors?

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[obtuse fog]

Do you mean whether B keeps the span the same?
If so, no. Consider a linear subspace L = ℒ({1, 1}, {2, 2}), so basically just ℒ({1, 1}). Apply a rotation matrix to it. Clearly, it won't be the same subspace.

Or did you mean that the image of the span is the same as the span of the images? If so, I think that's true... Though, I'm not 100% sure.

[distant lantern]

The question may require some clarification here

For instance, in a vector space of dimension 1, any linear transformation is a homothety

so in that sense, no matter what the linear transformation is, the resultants will stay in that line

likewise in a vector space of dimension 0

but in higher dimensions, there is no reason why a linear transformation should generally preserve a (strict) subspace

[dark marsh]

So if the span(v1,v2) where it is a linear line that you can graph it as so. But when you set Ax = b, where you graph b, it shows that it doesn't lie on the same line that of span(v1,v2).

bear with me because im like really early into linear algebra

[distant lantern]

first of all, if v1 and v2 are nonzero collinear vectors, then span(v1, v2)=span(v1)

second, I do not think you clarified the points I stated before

if your entire vector space is just a flat line, it is trivially true that any linear map is just a multiplication by a constant

in which case it still maps any element on the line, back on the line

[dark marsh]

so if im talking about the span(v1) i know that its a line, but the line can exist anywhere on the plane and that theres no "strict subspace" for it as you said before?

[distant lantern]

well, my point is whether the entire vector space you're considering was a line, a plane, a 3D space or anything bigger

no one says your vector spaces have to be of dimension 2

which is what I wanted to clarify with you

[dark marsh]

okay so, if the vector space is just 2 dimensions is what youre asking?

[distant lantern]

What is the dimension of your ambient vector space?

[dark marsh]

in 2d

[distant lantern]

so that is one q answered

So yes then Darpinger's answer applies

There is generally no reason for linear maps to preserve lines

see rotations for example

[dark marsh]

what happens if it was another dimensions like R^3

[distant lantern]

same thing

there is no reason for any linear transformation to preserve any line

in the eventuality there is one such line that is stable via this transformation, this line is an eigenspace of said transformation

[dark marsh]

okay what if i were to say to graph the span of the columns of the matrix. Would the span still be non-preservable on linear maps?

[distant lantern]

now that's an entirely different question

the span of multiple vectors may be a vector space of dimension bigger than 1

so it may not be a line anymore

in particular, for an invertible matrix, the columns span the entire vector space

so it would be hard to say that the span isn't preserved, it is trivially preserved

[dark marsh]

okay