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... $$Looking at the coordinates in the first position, this implies that:$$ 4x + 8y + 2z = 180  You can get two more similar equations by looking at the second and third coordinates. What are vector spaces and subspaces, rank, invertibility and more. So we multiply by this elimination matrix to our matrix A to form an Upper triangular matrix, U. This equation is pretty easy to solve. So we apply elimination matrices to change A \rightarrow U, where U is an upper traingular matrix. more cases of this system. So for any two vectors, \mathbf{w} and \mathbf{v}, a linear combination is: for any n vectors, \mathbf{v}_1,\mathbf{v}_2,\dots a linear combination will be: We will talk about what all these linear combinations for all c and d represent, later. There are infinite points on it but a special point (2,-1) also satisfies it and is on the line. First row of resultant matrix: 1 \times \text{row1} (of original matrix) + 0 \times \text{row2} + 0 \times \text{row3} = \text{row1}, Similarly second row of resultant matrix: -2 \times \text{row1} + 1 \times \text{row2} + 0 \times \text{row3} = \text{row2} - (2)\ \text{row1}. Now if we have two equations and we have to find solutions(s) that satisfy both these equations. e.g: a vector containing 2 and 3 is represented as: A column vector(or a row vector) can also be represented by paranthesis. We will make another Elimination matrix, E2 to perform this operation. It can occur between two vectors of same dimensions. And the vector \mathbf{b} = \begin{bmatrix} 7 \\3 \end{bmatrix} does not lie on that line and so no combination can do it. Vectors, Linear Combinations, Eliminations. Let’s talk about when there is more or less than one solution. [CDATA[ So subtracting 2 times the first equation from second and -1 times the first equation from third. Let’s see an example of a unique solution case. I need to generate the combinations of elements of two arrays with different lengths. How can I obtain the vector C containing sum of all the possible combinations of elements of … Now based on the coefficients, a system of equations may have a unique solution, an infinite number of solution, or no solution. Now in row3 we want to eliminate 10 using pivot -5, so we want to subtract -2 times the row 2 from row 3 which is also adding 2 times row 2 to row 3: so row3 of elimination matrix = % 