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basis: like the x,y,z axis but any dimension and they don't have to be all at perfect right angles to each other (that's an orthogonal basis)
coordinate: the numbers that tell you where you are on the basis. If you are at point (1,0,2) its like you travel 1 x axis, 0 y axis and 2 z axis which creates a line from the origin to your end point which can also be used as an axis, but this line can also be interpreted as just a bunch of numbers rather than an axis, called coordinates
if you do a linear transformation you change your axis to a new set of axis, the eigenvectors are the axis whose directions don't get changed under this transformation and the eigenvalue is the scale factor of how much the size is changed (if the size is the same, thats an eigenvalue of 1)
