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I don't have a background in math; I do have a basic understanding of linear algebra though. I am trying to understand the next part from "Deep learning" book:

enter image description here

I don't understand what they are explaining in that paragraph. I played around with the formula and I got:

$$\alpha=0, \ \ \ z=y$$

$$\alpha=1, \ \ \ z=x$$

$$\alpha=2,\ \ \ z=2x-y$$

$$\alpha=-1,\ \ \ z=-x+2y$$

$$\alpha=-2, \ \ \ z=-2x+3y$$

I don't know where this is going... Also per the paragraph explanation, I am expecting a formula that includes b.

Unfortunately, given my lack of math I am only comfortable with graphical explanations or real life examples.

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    $\begingroup$ they are explaining that if there are two (i.e., more than one) solutions, then there are infinitely many solutions (which can be generated from the two solutions $\mathbb x$ and $\mathbb y$) $\endgroup$ – J. W. Tanner Dec 3 at 0:31
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    $\begingroup$ I assume (2.11) is something like "$Ax = b$"? $\endgroup$ – Eric Towers Dec 3 at 0:31
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    $\begingroup$ If $Ax=b$ and $Ay=b$ then $$Az=A(\alpha x+(1-\alpha)y)=\alpha Ax+(1-\alpha)Ay=\alpha b+(1-\alpha)b=b.$$ $\endgroup$ – A.Γ. Dec 3 at 0:32
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(Not enough reputation to comment.)

As a comment, since you prefer graphical explanations, you can think of it as saying something like: "if two (or more) lines (or planes) intersect at more than one point, then they intersect at infinitely many points". The lines (or planes) represent solutions to equations.

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  • $\begingroup$ I thought in 2d geometry two lines can intersect at only one point... How can two lines intersect at more than one point? $\endgroup$ – Chicago1988 Dec 3 at 10:37
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    $\begingroup$ Two lines can intersect at more than one point if they lie exactly one over another. In this case, they intersect at infinitely many points. If you imagine three planes in space, you can try to convince yourself that it is also impossible for the three planes to all intersect at more than one point, unless if they intersect at infinitely many points. $\endgroup$ – Math321 Dec 4 at 18:49

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