introduce

If we write a linear system of equationsIs of the form, A isThe matrix of omega, 0 is omegaIs the vector that has m components 0. IfAll of the vectors are zero.I’ll call it the 0 vector, and I’ll get the resultTrivial solutionTo put it in plain language is that none of the vectors are exerting any force, so the resulting vector is not moving in space.Trivial solutionThe point of existence is to illustrate the matrixAll of the vectors in. This is the key to understanding homogeneous equations. If the matrixWhat happens when you have vectors in the same direction? This is theTrivial solutionSee example 1 of linear algebra and its applications P56.

The sample

The augmented matrix of linear equations is simplified to step type. Because each vector of the matrix has three components, the solution set is in three dimensional space and can be visualized by solid geometry.

Solution set decomposition form

Notice the first and third columns of the matrix

If I were to draw these two vectors in a three-dimensional coordinate system, they would have the same direction except for their magnitude. And one hypothesis is inIt’s going to be to the right, and the other vector is going to be atTo the left, you combine the coefficients of the two vectors to get the 0 vector. The vector in the second column let’s call it 0

If you write it in words, the first column vector is atIf I move to the right on the axis, and the third column goes to the left, how do I get the combination of going back to the origin

Results:



Once one is identifiedI can getNontrivial solutionAs long as the magnitude of the first column is the magnitude of the third columnTimes PI, you can just combine two vectors to get the 0 vector

ifAnother coordinate system(notice the wording here) in the coordinate systemThe trivial solutionThat is, by the vector 1, 2A line of theta.

conclusion

I’m not going to worry about the 0 space and so on. The main problem is how to get itNontrivial solutionThe above example is the simplestNontrivial solutionFor the above example, the literal and geometric description is that at least two vectors in the matrix are in the same direction (the size can be different because it can pass throughThe problem is best explained by drawing the vectors of a matrix in three dimensions. This solution can be used to solve problems in higher dimensional space.

The core of linear correlation is described in vector form: whether there are vectors in the matrix with the same direction, and there must be linear correlation between vectors with the same direction. Linear dependence not only enables linear equations to have nontrivial solutions, but also multiplies the solution set of equations