Even knowing that one solution was 3, so one factor must be x- 3 you couldn't factor it??mr_coffee said:i know, i can't factor that and get a nice number, i'd have to use the quadtract equation, but that can't be right because the book got a nice answer of 3.
Read it carefully! It specifically says "an eigenvalue", not the eigenvalue. And immediately below states that there is another and solves for it.mr_coffee said:ohhh wow i suck hah, thank u so much! why did they only use [tex]\lambda = 3[/tex] when it can also equal 1?
This is what we witnessed today.Even knowing that one solution was 3, so one factor must be x- 3 you couldn't factor it??
Eigenvalues and eigenvectors are mathematical concepts that are used to understand the behavior of linear transformations. Eigenvalues are scalar values that represent the scaling factor of the eigenvectors, which are special vectors that do not change direction when the transformation is applied to them.
To find eigenvalues and eigenvectors, you need to solve the characteristic equation of the transformation. This involves finding the determinant of the transformation matrix and setting it equal to zero. The resulting solutions are the eigenvalues. Then, for each eigenvalue, you can find the corresponding eigenvector by solving the system of equations formed by substituting the eigenvalue into the transformation matrix.
Eigenvalues and eigenvectors have many applications in mathematics and science. They are used to understand the behavior of linear systems, to find the optimal solutions for optimization problems, and to simplify complex equations. They also have practical uses in fields such as physics, engineering, and computer science.
If the transformation is represented by an n x n matrix, then there will be n eigenvalues and n corresponding eigenvectors. However, if the matrix is not diagonalizable, there may be fewer distinct eigenvalues and corresponding eigenvectors. To ensure that you have found all the eigenvalues and eigenvectors, you can check that the sum of the multiplicities of the eigenvalues is equal to the dimension of the matrix.
Yes, there are many software programs and online calculators that can find eigenvalues and eigenvectors for you. However, it is important to understand the mathematical concepts behind them in order to correctly interpret the results and use them in practical applications. Additionally, some transformations may be too complex for software to handle, so it is important to know how to find eigenvalues and eigenvectors by hand as well.