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The code blocks are much like those that were explained above for LeastSquaresPractice_4.py, but it’s a little shorter. The f_i‘s are our outputs. This post covers solving a system of equations from math to complete code, and it’s VERY closely related to the matrix inversion post. If you know basic calculus rules such as partial derivatives and the chain rule, you can derive this on your own. \footnotesize{\bold{X^T X}} is a square matrix. The code below is stored in the repo for this post, and it’s name is LeastSquaresPractice_Using_SKLearn.py. The noisy inputs, the system itself, and the measurement methods cause errors in the data. I’d like to do that someday too, but if you can accept equation 3.7 at a high level, and understand the vector differences that we did above, you are in a good place for understanding this at a first pass. We will cover linear dependency soon too. This is great! There’s one other practice file called LeastSquaresPractice_5.py that imports preconditioned versions of the data from conditioned_data.py. There are times that we’d want an inverse matrix of a system for repeated uses of solving for X, but most of the time we simply need a single solution of X for a system of equations, and there is a method that allows us to solve directly for X where we don’t need to know the inverse of the system matrix. Let’s test all this with some simple toy examples first and then move onto one real example to make sure it all looks good conceptually and in real practice. Once a diagonal element becomes 1 and all other elements in-column with it are 0’s, that diagonal element is a pivot-position, and that column is a pivot-column. Simultaneous Equations Solver Python Tessshlo. AND we could have gone through a lot more linear algebra to prove equation 3.7 and more, but there is a serious amount of extra work to do that. Data Scientist, PhD multi-physics engineer, and python loving geek living in the United States. Where do we go from here? v0 = ps0,0 * rs0,0 + ps0,1 * rs0,1 + ps0,2 * rs0,2 + y(ps0,0 * v0 + ps0,1 * v1 + ps0,2 *v2) I am solving for v0,v1,v2. \footnotesize{\bold{X}} is \footnotesize{4x3} and it’s transpose is \footnotesize{3x4}. Example. Let’s put the above set of equations in matrix form (matrices and vectors will be bold and capitalized forms of their normal font lower case subscripted individual element counterparts). The programming (extra lines outputting documentation of steps have been deleted) is in the block below. Consider the next section if you want. Let’s remember that our objective is to find the least of the squares of the errors, which will yield a model that passes through the data with the least amount of squares of the errors. Next we enter the for loop for the fd‘s. Could we derive a least squares solution using the principles of linear algebra alone? 2y + 5z = -4. In the first code block, we are not importing our pure python tools. The error that we want to minimize is: This is why the method is called least squares. Here’s another convenience. Since we have two equations and two unknowns, we can find a unique solution for \footnotesize{\bold{W_1}}. Use the python programming enviroment to write a code can solve a system of linear equations with n variables by Guass jordan methods. There are other Jupyter notebooks in the GitHub repository that I believe you will want to look at and try out. When solving linear equations, we can represent them in matrix form. Doing row operations on A to drive it to an identity matrix, and performing those same row operations on B, will drive the elements of B to become the elements of X. We’ll use python again, and even though the code is similar, it is a bit differ… However, there is a way to find a \footnotesize{\bold{W^*}} that minimizes the error to \footnotesize{\bold{Y_2}} as \footnotesize{\bold{X_2 W^*}} passes thru the column space of \footnotesize{\bold{X_2}}. (row 3 of A_M) – 2.4 * (row 2 of A_M) (row 3 of B_M) – 2.4 * (row 2 of B_M), 7. The new set of equations would then be the following. We’ll use python again, and even though the code is similar, it is a bit different. Thus, equation 2.7b brought us to a point of being able to solve for a system of equations using what we’ve learned before. When we replace the \footnotesize{\hat{y}_i} with the rows of \footnotesize{\bold{X}} is when it becomes interesting. These substitutions are helpful in that they simplify all of our known quantities into single letters. We’ll even throw in some visualizations finally. IF you want more, I refer you to my favorite teacher (Sal Kahn), and his coverage on these linear algebra topics HERE at Khan Academy. Computes the “exact” solution, x, of the well-determined, i.e., full rank, linear matrix equation ax = b. We then used the test data to compare the pure python least squares tools to sklearn’s linear regression tool that used least squares, which, as you saw previously, matched to reasonable tolerances. Once we encode each text element to have it’s own column, where a “1” only occurs when the text element occurs for a record, and it has “0’s” everywhere else. Setting equation 1.10 to 0 gives. Again, to go through ALL the linear algebra for supporting this would require many posts on linear algebra. Develop libraries for array computing, recreating NumPy's foundational concepts. The only variables that we must keep visible after these substitutions are m and b. Since I have done this before, I am going to ask you to trust me with a simplification up front. numpy documentation: Solve linear systems with np.solve. Let’s cover the differences. (row 2 of A_M) – 0.472 * (row 3 of A_M) (row 2 of B_M) – 0.472 * (row 3 of B_M). Also, the train_test_split is a method from the sklearn modules to use most of our data for training and some for testing. Finally, let’s give names to our matrix and vectors. You’ll know when a bias in included in a system matrix, because one column (usually the first or last column) will be all 1’s. Posted By: Carlo Bazzo May 20, 2019. Solving linear equations using matrices and Python TOPICS: Analytics EN Python. Therefore, we want to find a reliable way to find m and b that will cause our line equation to pass through the data points with as little error as possible. Computes the “exact” solution, x, of the well-determined, i.e., full rank, linear matrix equation ax = b. Parameters a (…, M, M) array_like. They store almost all of the equations for this section in them. The solution method is a set of steps, S, focusing on one column at a time. Without using (import numpy) as np and (import sys) Our realistic data set was obtained from HERE. To do this you use the solve() command: >>> solution = sym. And that system has output data that can be measured. We will look at matrix form along with the equations written out as we go through this to keep all the steps perfectly clear for those that aren’t as versed in linear algebra (or those who know it, but have cold memories on it – don’t we all sometimes). numpy.linalg.solve¶ numpy.linalg.solve (a, b) [source] ¶ Solve a linear matrix equation, or system of linear scalar equations. Let’s look at the 3D output for this toy example in figure 3 below, which uses fake and well balanced output data for easy visualization of the least squares fitting concept. Let’s assume that we have a system of equations describing something we want to predict. I am also a fan of THIS REFERENCE. A file named LinearAlgebraPurePython.py contains everything needed to do all of this in pure python. Next is fitting polynomials using our least squares routine. If you’ve never been through the linear algebra proofs for what’s coming below, think of this at a very high level. Let’s examine that using the next code block below. This work could be accomplished in as few as 10 – 12 lines of python. When the dimensionality of our problem goes beyond two input variables, just remember that we are now seeking solutions to a space that is difficult, or usually impossible, to visualize, but that the values in each column of our system matrix, like \footnotesize{\bold{A_1}}, represent the full record of values for each dimension of our system including the bias (y intercept or output value when all inputs are 0). It’s a worthy study though. If our set of linear equations has constraints that are deterministic, we can represent the problem as matrices and apply matrix algebra. It’s hours long, but worth the investment. 1/5.0 * (row 1 of A_M) and 1/5.0 * (row 1 of B_M), 2. Thus, if we transform the left side of equation 3.8 into the null space using \footnotesize{\bold{X_2^T}}, we can set the result equal to the zero vector (we transform into the null space), which is represented by equation 3.9. However, there is an even greater advantage here. Let’s do similar steps for \frac{\partial E}{\partial b} by setting equation 1.12 to “0”. Let’s look at the output from the above block of code. (row 2 of A_M) – 3.0 * (row 1 of A_M) (row 2 of B_M) – 3.0 * (row 1 of B_M), 3. We now do similar operations to find m. Let’s multiply equation 1.15 by N and equation 1.16 by U and subtract the later from the former as shown next. There are complementary .py files of each notebook if you don’t use Jupyter. This blog’s work of exploring how to make the tools ourselves IS insightful for sure, BUT it also makes one appreciate all of those great open source machine learning tools out there for Python (and spark, and there’s ones for R of course, too). 1/3.667 * (row 3 of A_M) and 1/3.667 * (row 3 of B_M), 8. Second, multiply the transpose of the input data matrix onto the input data matrix. In case you weren’t aware, when we multiply one matrix on another, this transforms the right matrix into the space of the left matrix. Then, for each row without fd in them, we: We do those steps for each row that does not have the focus diagonal in it to drive all the elements in the current column to 0 that are NOT in the row with the focus diagonal in it. The documentation for numpy.linalg.solve (that’s the linear algebra solver of numpy) is HERE. Gaining greater insight into machine learning tools is also quite enhanced thru the study of linear algebra. You don’t even need least squares to do this one. We can isolate b by multiplying equation 1.15 by U and 1.16 by T and then subtracting the later from the former as shown next. Now, here is the trick. OK. That worked, but will it work for more than one set of inputs? As always, I encourage you to try to do as much of this on your own, but peek as much as you want for help. The fewest lines of code are rarely good code. ... Systems of linear equations. (row 1 of A_M) – 0.6 * (row 2 of A_M) (row 1 of BM) – 0.6 * (row 2 of B_M), 6. numpy.linalg.solve¶ linalg.solve (a, b) [source] ¶ Solve a linear matrix equation, or system of linear scalar equations. The APMonitor Modeling Language with a Python interface is optimization software for mixed-integer and differential algebraic equations. Then just return those coefficients for use. We then operate on the remaining rows, the ones without fd in them, as follows: We do this for columns from left to right in both the A and B matrices. Understanding the derivation is still better than not seeking to understand it. Instead, we are importing the LinearRegression class from the sklearn.linear_model module. Let’s recap where we’ve come from (in order of need, but not in chronological order) to get to this point with our own tools: We’ll be using the tools developed in those posts, and the tools from those posts will make our coding work in this post quite minimal and easy. Let’s walk through this code and then look at the output. Therefore, B_M morphed into X. Section 2 is further making sure that our data is formatted appropriately – we want more rows than columns. Why do we focus on the derivation for least squares like this? Let’s use equation 3.7 on the right side of equation 3.6. We now have closed form solutions for m and b that will draw a line through our points with minimal error between the predicted points and the measured points. Section 4 is where the machine learning is performed. Using the steps illustrated in the S matrix above, let’s start moving through the steps to solve for X. Check out the operation if you like. Then we algebraically isolate m as shown next. Now let’s use those shorthanded methods above to simplify equations 1.19 and 1.20 down to equations 1.21 and 1.22. Both sides of equation 3.4 are in our column space. The first nested for loop works on all the rows of A besides the one holding fd. I’ll try to get those posts out ASAP. Our starting matrices, A and B, are copied, code wise, to A_M and B_M to preserve A and B for later use. Python's numerical library NumPy has a function numpy.linalg.solve() which solves a linear matrix equation, or system of linear scalar equation. However, it’s a testimony to python that solving a system of equations could be done with so little code. Let’s look at the dimensions of the terms in equation 2.7a remembering that in order to multiply two matrices or a matrix and a vector, the inner dimensions must be the same (e.g. Now, let’s subtract \footnotesize{\bold{Y_2}} from both sides of equation 3.4. If you’ve been through the other blog posts and played with the code (and even made it your own, which I hope you have done), this part of the blog post will seem fun. Now here’s a spoiler alert. If you learned and understood, you are well on your way to being able to do such things from scratch once you’ve learned the math for future algorithms. Published by Thom Ives on December 3, 2018December 3, 2018, Find the complimentary System Of Equations project on GitHub. If we stretch the spring to integral values of our distance unit, we would have the following data points: Hooke’s law is essentially the equation of a line and is the application of linear regression to the data associated with force, spring displacement, and spring stiffness (spring stiffness is the inverse of spring compliance). Python solve linear equations you solving a system of in pure without numpy or scipy integrated machine learning and artificial intelligence with gaussian elimination martin thoma solved the following set using s chegg com algebra w symbolic maths tutorial linux hint systems Python Solve Linear Equations You Solving A System Of Equations In Pure Python Without Numpy Or… Read More » Block 3 does the actual fit of the data and prints the resulting coefficients for the model. Both of these files are in the repo. Solving Ordinary Diffeial Equations. The block structure follows the same structure as before, but, we are using two sets of input data now. I hope the amount that is presented in this post will feel adequate for our task and will give you some valuable insights. Now we want to find a solution for m and b that minimizes the error defined by equations 1.5 and 1.6. Those previous posts were essential for this post and the upcoming posts. When this is complete, A is an identity matrix, and B has become the solution for X. It could be done without doing this, but it would simply be more work, and the same solution is achieved more simply with this simplification. To understand and gain insights. The matrix rank will tell us that. There are times that we’d want an inverse matrix of a system for repeated uses of solving for X, but most of the time we simply need a single solution of X for a system of equations, and there is a method that allows us to solve directly for Xwhere we don’t need to know the inverse of the system matrix. How to do gradient descent in python without numpy or scipy. Solves systems of linear equations. This file is in the repo for this post and is named LeastSquaresPractice_4.py. If we repeat the above operations for all \frac{\partial E}{\partial w_j} = 0, we have the following. Now, let’s arrange equations 3.1a into matrix and vector formats. Click on the appropriate link for additional information and source code. I hope you’ll run the code for practice and check that you got the same output as me, which is elements of X being all 1’s. 1/7.2 * (row 2 of A_M) and 1/7.2 * (row 2 of B_M), 5. We have not yet covered encoding text data, but please feel free to explore the two functions included in the text block below that does that encoding very simply. We’ll then learn how to use this to fit curved surfaces, which has some great applications on the boundary between machine learning and system modeling and other cool/weird stuff. How to do gradient descent in python without numpy or scipy. Considering the operations in equation 2.7a, the left and right both have dimensions for our example of \footnotesize{3x1}. Linear and nonlinear equations can also be solved with Excel and MATLAB. Please note that these steps focus on the element used for scaling within the current row operations. Understanding this will be very important to discussions in upcoming posts when all the dimensions are not necessarily independent, and then we need to find ways to constructively eliminate input columns that are not independent from one of more of the other columns. Where \footnotesize{\bold{F}} and \footnotesize{\bold{W}} are column vectors, and \footnotesize{\bold{X}} is a non-square matrix. As you’ve seen above, we were comparing our results to predictions from the sklearn module. Wikipedia defines a system of linear equationsas: The ultimate goal of solving a system of linear equations is to find the values of the unknown variables. That is …. B has been renamed to B_M, and the elements of B have been renamed to b_m, and the M and m stand for morphed, because with each step, we are changing (morphing) the values of B. Published by Thom Ives on December 16, 2018December 16, 2018. For the number “n” of related encoded columns, we always have “n-1” columns, and the case where the two elements we use are both “0” is the case where the nth element would exist. Let’s start with single input linear regression. Let’s create some short handed versions of some of our terms. There are multiple ways to solve such a system, such as Elimination of Variables, Cramer's Rule, Row Reduction Technique, and the Matrix Solution. The code in python employing these methods is shown in a Jupyter notebook called SystemOfEquationsStepByStep.ipynb in the repo. Third, front multiply the transpose of the input data matrix onto the output data matrix. The mathematical convenience of this will become more apparent as we progress. Also, we know that numpy or scipy or sklearn modules could be used, but we want to see how to solve for X in a system of equations without using any of them, because this post, like most posts on this site, is about understanding the principles from math to complete code. With the tools created in the previous posts (chronologically speaking), we’re finally at a point to discuss our first serious machine learning tool starting from the foundational linear algebra all the way to complete python code. Now let’s use the chain rule on E using a also. SymPy is written entirely in Python and does not require any external libraries. (row 1 of A_M) – -0.083 * (row 3 of A_M) (row 1 of B_M) – -0.083 * (row 3 of B_M), 9. \footnotesize{\bold{W}} is \footnotesize{3x1}. Please appreciate that I completely contrived the numbers, so that we’d come up with an X of all 1’s. After reviewing the code below, you will see that sections 1 thru 3 merely prepare the incoming data to be in the right format for the least squares steps in section 4, which is merely 4 lines of code. However, IF we were to cover all the linear algebra required to understand a pure linear algebraic derivation for least squares like the one below, we’d need a small textbook on linear algebra to do so. If we used the nth column, we’d create a linear dependency (colinearity), and then our columns for the encoded variables would not be orthogonal as discussed in the previous post. This tutorial is an introduction to solving linear equations with Python. That’s just two points. Realize that we went through all that just to show why we could get away with multiplying both sides of the lower left equation in equations 3.2 by \footnotesize{\bold{X_2^T}}, like we just did above in the lower equation of equations 3.9, to change the not equal in equations 3.2 to an equal sign? And to make the denominator match that of equation 1.17, we simply multiply the above equation by 1 in the form of \frac{-1}{-1}. A \cdot B_M should be B and it is! Pycse Python3 Comtions In Science And Engineering. Start from the left column and moving right, we name the current diagonal element the focus diagonal (fd) element. In this post, we create a clustering algorithm class that uses the same principles as scipy, or sklearn, but without using sklearn or numpy or scipy. In a previous article, we looked at solving an LP problem, i.e. At the top portion of the code, copies of A and B are saved for later use, and we save A‘s square dimension for later use. Then, like before, we use pandas features to get the data into a dataframe and convert that into numpy versions of our X and Y data. That is, we have more equations than unknowns, and therefore \footnotesize{ \bold{X}} has more rows than columns. These steps are essentially identical to the steps presented in the matrix inversion post. Computes the “exact” solution, x, of the well-determined, i.e., full rank, linear matrix equation ax = b. Here is an example of a system of linear equations with two unknown variables, x and y: Equation 1: To solve the above system of linear equations, we need to find the values of the x and yvariables. Let’s consider the parts of the equation to the right of the summation separately for a moment. Is there yet another way to derive a least squares solution? We still want to minimize the same error as was shown above in equation 1.5, which is repeated here next. Applying Polynomial Features to Least Squares Regression using Pure Python without Numpy or Scipy, \tag{1.3} x=0, \,\,\,\,\, F = k \cdot 0 + F_b \\ x=1, \,\,\,\,\, F = k \cdot 1 + F_b \\ x=2, \,\,\,\,\, F = k \cdot 2 + F_b, \tag{1.5} E=\sum_{i=1}^N \lparen y_i - \hat y_i \rparen ^ 2, \tag{1.6} E=\sum_{i=1}^N \lparen y_i - \lparen mx_i+b \rparen \rparen ^ 2, \tag{1.7} a= \lparen y_i - \lparen mx_i+b \rparen \rparen ^ 2, \tag{1.8} \frac{\partial E}{\partial a} = 2 \sum_{i=1}^N \lparen y_i - \lparen mx_i+b \rparen \rparen, \tag{1.9} \frac{\partial a}{\partial m} = -x_i, \tag{1.10} \frac{\partial E}{\partial m} = \frac{\partial E}{\partial a} \frac{\partial a}{\partial m} = 2 \sum_{i=1}^N \lparen y_i - \lparen mx_i+b \rparen \rparen \lparen -x_i \rparen), \tag{1.11} \frac{\partial a}{\partial b} = -1, \tag{1.12} \frac{\partial E}{\partial b} = \frac{\partial E}{\partial a} \frac{\partial a}{\partial b} = 2 \sum_{i=1}^N \lparen y_i - \lparen mx_i+b \rparen \rparen \lparen -1 \rparen), 0 = 2 \sum_{i=1}^N \lparen y_i - \lparen mx_i+b \rparen \rparen \lparen -x_i \rparen), 0 = \sum_{i=1}^N \lparen -y_i x_i + m x_i^2 + b x_i \rparen), 0 = \sum_{i=1}^N -y_i x_i + \sum_{i=1}^N m x_i^2 + \sum_{i=1}^N b x_i, \tag{1.13} \sum_{i=1}^N y_i x_i = \sum_{i=1}^N m x_i^2 + \sum_{i=1}^N b x_i, 0 = 2 \sum_{i=1}^N \lparen -y_i + \lparen mx_i+b \rparen \rparen, 0 = \sum_{i=1}^N -y_i + m \sum_{i=1}^N x_i + b \sum_{i=1} 1, \tag{1.14} \sum_{i=1}^N y_i = m \sum_{i=1}^N x_i + N b, T = \sum_{i=1}^N x_i^2, \,\,\, U = \sum_{i=1}^N x_i, \,\,\, V = \sum_{i=1}^N y_i x_i, \,\,\, W = \sum_{i=1}^N y_i, \begin{alignedat} ~&mTU + bU^2 &= &~VU \\ -&mTU - bNT &= &-WT \\ \hline \\ &b \lparen U^2 - NT \rparen &= &~VU - WT \end{alignedat}, \begin{alignedat} ~&mNT + bUN &= &~VN \\ -&mU^2 - bUN &= &-WU \\ \hline \\ &m \lparen TN - U^2 \rparen &= &~VN - WU \end{alignedat}, \tag{1.18} m = \frac{-1}{-1} \frac {VN - WU} {TN - U^2} = \frac {WU - VN} {U^2 - TN}, \tag{1.19} m = \dfrac{\sum\limits_{i=1}^N x_i \sum\limits_{i=1}^N y_i - N \sum\limits_{i=1}^N x_i y_i}{ \lparen \sum\limits_{i=1}^N x_i \rparen ^2 - N \sum\limits_{i=1}^N x_i^2 }, \tag{1.20} b = \dfrac{\sum\limits_{i=1}^N x_i y_i \sum\limits_{i=1}^N x_i - N \sum\limits_{i=1}^N y_i \sum\limits_{i=1}^N x_i^2 }{ \lparen \sum\limits_{i=1}^N x_i \rparen ^2 - N \sum\limits_{i=1}^N x_i^2 }, \overline{x} = \frac{1}{N} \sum_{i=1}^N x_i, \,\,\,\,\,\,\, \overline{xy} = \frac{1}{N} \sum_{i=1}^N x_i y_i, \tag{1.21} m = \frac{N^2 \overline{x} ~ \overline{y} - N^2 \overline{xy} } {N^2 \overline{x}^2 - N^2 \overline{x^2} } = \frac{\overline{x} ~ \overline{y} - \overline{xy} } {\overline{x}^2 - \overline{x^2} }, \tag{1.22} b = \frac{\overline{xy} ~ \overline{x} - \overline{y} ~ \overline{x^2} } {\overline{x}^2 - \overline{x^2} }, \tag{Equations 2.1} f_1 = x_{11} ~ w_1 + x_{12} ~ w_2 + b \\ f_2 = x_{21} ~ w_1 + x_{22} ~ w_2 + b \\ f_3 = x_{31} ~ w_1 + x_{32} ~ w_2 + b \\ f_4 = x_{41} ~ w_1 + x_{42} ~ w_2 + b, \tag{Equations 2.2} f_1 = x_{10} ~ w_0 + x_{11} ~ w_1 + x_{12} ~ w_2 \\ f_2 = x_{20} ~ w_0 + x_{21} ~ w_1 + x_{22} ~ w_2 \\ f_3 = x_{30} ~ w_0 + x_{31} ~ w_1 + x_{32} ~ w_2 \\ f_4 = x_{40} ~ w_0 + x_{41} ~ w_1 + x_{42} ~ w_2, \tag{2.3} \bold{F = X W} \,\,\, or \,\,\, \bold{Y = X W}, \tag{2.4} E=\sum_{i=1}^N \lparen y_i - \hat y_i \rparen ^ 2 = \sum_{i=1}^N \lparen y_i - x_i ~ \bold{W} \rparen ^ 2, \tag{Equations 2.5} \frac{\partial E}{\partial w_j} = 2 \sum_{i=1}^N \lparen y_i - x_i \bold{W} \rparen \lparen -x_{ij} \rparen = 2 \sum_{i=1}^N \lparen f_i - x_i \bold{W} \rparen \lparen -x_{ij} \rparen \\ ~ \\ or~using~just~w_1~for~example \\ ~ \\ \begin{alignedat}{1} \frac{\partial E}{\partial w_1} &= 2 \lparen f_1 - \lparen x_{10} ~ w_0 + x_{11} ~ w_1 + x_{12} ~ w_2 \rparen \rparen x_{11} \\ &+ 2 \lparen f_2 - \lparen x_{20} ~ w_0 + x_{21} ~ w_1 + x_{22} ~ w_2 \rparen \rparen x_{21} \\ &+ 2 \lparen f_3 - \lparen x_{30} ~ w_0 + x_{31} ~ w_1 + x_{32} ~ w_2 \rparen \rparen x_{31} \\ &+ 2 \lparen f_4 - \lparen x_{40} ~ w_0 + x_{41} ~ w_1 + x_{42} ~ w_2 \rparen \rparen x_{41} \end{alignedat}, \tag{2.6} 0 = 2 \sum_{i=1}^N \lparen y_i - x_i \bold{W} \rparen \lparen -x_{ij} \rparen, \,\,\,\,\, \sum_{i=1}^N y_i x_{ij} = \sum_{i=1}^N x_i \bold{W} x_{ij} \\ ~ \\ or~using~just~w_1~for~example \\ ~ \\ f_1 x_{11} + f_2 x_{21} + f_3 x_{31} + f_4 x_{41} \\ = \left( x_{10} ~ w_0 + x_{11} ~ w_1 + x_{12} ~ w_2 \right) x_{11} \\ + \left( x_{20} ~ w_0 + x_{21} ~ w_1 + x_{22} ~ w_2 \right) x_{21} \\ + \left( x_{30} ~ w_0 + x_{31} ~ w_1 + x_{32} ~ w_2 \right) x_{31} \\ + \left( x_{40} ~ w_0 + x_{41} ~ w_1 + x_{42} ~ w_2 \right) x_{41} \\ ~ \\ the~above~in~matrix~form~is \\ ~ \\ \bold{ X_j^T Y = X_j^T F = X_j^T X W}, \tag{2.7b} \bold{ \left(X^T X \right) W = \left(X^T Y \right)}, \tag{3.1a}m_1 x_1 + b_1 = y_1\\m_1 x_2 + b_1 = y_2, \tag{3.1b} \begin{bmatrix}x_1 & 1 \\ x_2 & 1 \end{bmatrix} \begin{bmatrix}m_1 \\ b_1 \end{bmatrix} = \begin{bmatrix}y_1 \\ y_2 \end{bmatrix}, \tag{3.1c} \bold{X_1} = \begin{bmatrix}x_1 & 1 \\ x_2 & 1 \end{bmatrix}, \,\,\, \bold{W_1} = \begin{bmatrix}m_1 \\ b_1 \end{bmatrix}, \,\,\, \bold{Y_1} = \begin{bmatrix}y_1 \\ y_2 \end{bmatrix}, \tag{3.1d} \bold{X_1 W_1 = Y_1}, \,\,\, where~ \bold{Y_1} \isin \bold{X_{1~ column~space}}, \tag{3.2a}m_2 x_1 + b_2 = y_1 \\ m_2 x_2 + b_2 = y_2 \\ m_2 x_3 + b_2 = y_3 \\ m_2 x_4 + b_2 = y_4, \tag{3.1b} \begin{bmatrix}x_1 & 1 \\ x_2 & 1 \\ x_3 & 1 \\ x_4 & 1 \end{bmatrix} \begin{bmatrix}m_2 \\ b_2 \end{bmatrix} = \begin{bmatrix}y_1 \\ y_2 \\ y_3 \\ y_4 \end{bmatrix}, \tag{3.2c} \bold{X_2} = \begin{bmatrix}x_1 & 1 \\ x_2 & 1 \\ x_3 & 1 \\ x_4 & 1 \end{bmatrix}, \,\,\, \bold{W_2} = \begin{bmatrix}m_2 \\ b_2 \end{bmatrix}, \,\,\, \bold{Y_2} = \begin{bmatrix}y_1 \\ y_2 \\ y_3 \\ y_4 \end{bmatrix}, \tag{3.2d} \bold{X_2 W_2 = Y_2}, \,\,\, where~ \bold{Y_2} \notin \bold{X_{2~ column~space}}, \tag{3.4} \bold{X_2 W_2^* = proj_{C_s (X_2)}( Y_2 )}, \tag{3.5} \bold{X_2 W_2^* - Y_2 = proj_{C_s (X_2)} (Y_2) - Y_2}, \tag{3.6} \bold{X_2 W_2^* - Y_2 \isin C_s (X_2) ^{\perp} }, \tag{3.7} \bold{C_s (A) ^{\perp} = N(A^T) }, \tag{3.8} \bold{X_2 W_2^* - Y_2 \isin N (X_2^T) }, \tag{3.9} \bold{X_2^T X_2 W_2^* - X_2^T Y_2 = 0} \\ ~ \\ \bold{X_2^T X_2 W_2^* = X_2^T Y_2 }, BASIC Linear Algebra Tools in Pure Python without Numpy or Scipy, Find the Determinant of a Matrix with Pure Python without Numpy or Scipy, Simple Matrix Inversion in Pure Python without Numpy or Scipy, Solving a System of Equations in Pure Python without Numpy or Scipy, Gradient Descent Using Pure Python without Numpy or Scipy, Clustering using Pure Python without Numpy or Scipy, Least Squares with Polynomial Features Fit using Pure Python without Numpy or Scipy, Single Input Linear Regression Using Calculus, Multiple Input Linear Regression Using Calculus, Multiple Input Linear Regression Using Linear Algebraic Principles.

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