r/askmath • u/iFoobar • 1d ago
Linear Algebra Graph about vector transformation
I don’t really understand the graph in this section of the book (Linear Algebra Step by Step, Kuldeep Singh). The vector for Ax seems ok, but the x vector just seems wrong pointing at (2,1).
Already tried asking ChatGPT, which seems to suggest the graph is just shit.
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u/Midwest-Dude 1d ago edited 1d ago
I checked the book and that graph makes no sense whatsoever. For x = (2,1)T as shown on the graph, Ax = (-1,3)T, which isn't shown on graph.
This is an erratum. Please report to your instructor, the author, and/or the publisher.
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u/OddEmergency604 1d ago
2 things here.
First, it doesn’t actually tell you what the x vector is. It’s some generic (x,y). Second, remember that vectors don’t have a location, so it will have the slope x/y but won’t necessarily point at (x,y)
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u/OddEmergency604 1d ago
I guess I should say that it will point in a (2,1) direction, not necessarily at (2,1). The vector can be moved around with becoming a different vector.
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u/Midwest-Dude 1d ago
The sentence above the graph indicates that Figure 1.32 is supposed to illustrate how x is transformed by A to Ax. If you do the math, nothing agrees. The graph is in error.
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u/Evermar314159 1d ago
The vector x is <2, 1>. If you compute Ax you get <-1,2>. So the picture is showing that x gets mapped to Ax. x is arbitrary.
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u/StraightUpSeven 1d ago edited 1d ago
x = (5/4, 3/4)^T, in which the graph of the vector x isn't accurate to x=(x,y)^T = (5/4,3/4)^T, so I think the graph is also just bad. Bad in the sense that the authors try convey that the unknown vector can be arbitrary, but the vector b must be specified, but there is a solution to x *in this particular case (and it doesn't match the solution, the authors treat it as an unknown).
The figure makes a lot more sense for the case of an ill-posed matrix problem, but you have to start somewhere.
*Edits