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 Mar 3 comment All possible solutions to the Matrix Equation (free variables appearing) Your application of Solve gives a set of solutions with four free parameters, x[9], x[10], x[11], and x[12]. That's the complete set of all solutions. Mar 3 comment All possible solutions to the Matrix Equation (free variables appearing) @VF1 By not stipulating any values for x[9] through x[12], Solve really is returning all solutions. Compare its output where the null space is a little more complicated to represent, as in m = Array[1 &, {3, 3}]; Block[{var = Array[x, Last[Dimensions[m]]]}, Solve[m.var == {0, 0, 0}, var]]. Even without MaxExtraConditions, and despite issuing a warning, Solve has obtained all solutions. Mar 3 revised Plot a 2D vector path onto a surface added 108 characters in body Mar 3 comment Plot a 2D vector path onto a surface +1 This does it in a nutshell. One possible improvement would be not to double-compute p[t] in ParametricPlot3D, anticipating applications where this could be an expensive function. That's not an issue for this example, of course. I wonder, though, whether a little more explanation of how this solution was developed--and a less condensed presentation of the code--might not be more appropriate for this question given the stated background and needs of the O.P. :-). Mar 3 comment All possible solutions to the Matrix Equation (free variables appearing) Pavithran, the null space is critical to finding all solutions, because every solution can be obtained as any particular solution (where the RHS is not necessary zero) plus some element of the null space. If this is not perfectly clear, please consult any textbook on linear algebra. Mar 3 answered Plot a 2D vector path onto a surface Mar 3 revised Plot a 2D vector path onto a surface added 11 characters in body; edited tags Mar 3 reviewed Approve All possible solutions to the Matrix Equation (free variables appearing) Mar 3 comment All possible solutions to the Matrix Equation (free variables appearing) You're right; I tried exactly that but introduced a typo. I stand corrected. Have you looked into NullSpace? It can be used to produce all solutions to LinearSolve once you have a single one. E.g., test it with InpMatrix.({0, 0, 0, 0, 0, 0, 1, -45, 0, 0, 0, 0} + {x1, x2, x3, x4}.NullSpace[InpMatrix]) Notice that the output of NullSpace shows in this case that you can freely vary the last four coefficients. Mar 3 comment All possible solutions to the Matrix Equation (free variables appearing) The "solution you are looking for" clearly is not a solution: when you left-multiply it by InpMatrix, you do not obtain synd. Have you perhaps mixed up the roles of the solution and synd? Mar 3 comment How to get the matrix coefficient from a QuadraticForm expression in a very fast way? This is indeed very fast, but it does not return correct values. Mar 3 answered How to get the matrix coefficient from a QuadraticForm expression in a very fast way? Mar 1 awarded Scholar Mar 1 accepted Strategies for solving problems involving searches Mar 1 comment How do you draw credible regions/intervals on a 2D PDF? @PlatoManiac I think your answer is nice and will be helpful for future visitors; not only that, it's the accepted one! Please don't delete it. I have been commenting on it only out of genuine curiosity and a desire to clarify which replies truly answer which interpretations of the question. To all others: (1) thank you for your kind recognition; (2) the achievement of high reputations here is a mark of this community's generosity and understanding of how SE works; (3) all comments about rep are understood to be both friendly and to have traces of levity--especially "firmly in 2nd place." :-) Mar 1 comment How do you draw credible regions/intervals on a 2D PDF? It would be good to know what these are "confidence regions" of. They are not confidence regions of the mean, mode, or median, for instance: such regions would be tremendously smaller in extent than shown here. To be clear, I'm not questioning your beautiful work, but I am asking for clarification of what it is intended to mean and how it is supposed to be interpreted. Mar 1 answered How do you draw credible regions/intervals on a 2D PDF? Mar 1 comment How do you draw credible regions/intervals on a 2D PDF? The usual terminology for this is a 90% credible region, Jay. There typically exist infinitely many such regions in multivariate situations, so it helps to provide some constraints, such as asking for a credible region of the smallest diameter that encloses the mode (if the parameters are comparable to one another), for example, or to insist that the credible region be bounded by an isocontour of the density. Mar 1 comment How do you draw credible regions/intervals on a 2D PDF? NB: This is a confidence region for the mean, not for the PDF. I have asked the OP for clarification. (A CI for the PDF would be a pair of surfaces enclosing the PDF; it could not be depicted in any clear fashion with a contour plot or array plot.) Mar 1 comment How do you draw credible regions/intervals on a 2D PDF? Jay, what sort of "confidence intervals" do you want? The most natural ones would be confidence intervals for the density estimates themselves: that is, confidence intervals for values of the PDF. However, Platomaniac's answer seems to think you intended confidence intervals for the mean. The two are completely different things, so please clarify your intentions.