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Suppose, there is a system of linear equations $Ax = b$, where $A \in R^{m \times n}, m>n$ and $b_i \in \{l_1,....,l_k\}$, i.e. each $b_i$ is a label instead of being a real number.

To solve this, we can regroup the left hand sides which have the same label and we have such sets of such equations for each label. I think this would give us an under-determined system of equations. How would one solve it?

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    What are the entries of $A$, and what dimensions should $x$ have?2012-11-02
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    Usually with categorical variables, one category is chosen as a baseline and all others are assigned their own indicator variable in the regression. For example, if the original variable represents weight and the categories are "low", "medium" and "high", we may choose "medium" as the baseline and create two other variables named "isLow" and "isHigh". If a person has "medium" weight, isLow and isHigh will both be zero. If a person has "low" weight, isLow will be one and isHigh will be zero. If a person has "high" weight, isLow will be zero and isHigh will be one.2012-11-02
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    @HansEngler $A$ has real entries and $x$ is an $n \times 1$ vector.2012-11-02
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    @Max I get what you mean. But by assigning numerical values to the labels based on what they mean, we are forcing the solution $x$ to be in the range of these assigned values. I was looking for ways where we just use the label information from the RHS and solve the derived LHS equations.2012-11-02
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    @locke14 I misunderstood the question. My response was for $A$ having these entries, not $b$.2012-11-02
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    What exactly is the meaning of these labels? Are they just for forming subsets of equations? If so, what's the intended relationship between the various subsets? I think you'll have to explain the setup in a bit more detail.2012-11-05
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    @locke14: If $A$ is a real matrix and $x$ is a real vector, then $b$ is a vector of real numbers. But if the entries of $x$ are somehow also labels, you have to explain what it means to multiply them by real numbers. For example, if the labels are "U", "V", "W", what does it mean to compute $1.5 \cdot U + \sqrt{3} \cdot V - W$? Such expressions would occur in the matrix product.2012-11-16

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