Linear dependence, independence, and singular and non-sinular matrices
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Linear dependence, independence, and singular and non-singular matrices
Compare the following 2 ´ 2 matrices: [Equation goes here - download the original to see it.] But we cannot divide by 0; hence this operation for this matrix is ill-defined and A cannot have an inverse. The determinant of a matrix is [Equation goes here - download the original to see it.] In the case of [Equation goes here - download the original to see it.] , A does not have an inverse because det A = 0. Matrices for which det A = 0 are called singular matrices. When the matrix is called non-singular. In general, for any square matrix [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] There is another feature of A that is linked to the reason why A does not have an inverse.[Equation goes here - download the original to see it.] Notice that the second row is twice the first row. For matrices of any size, when we multiply and add rows (or columns) of a matrix, then we are forming a linear combination of them. When one row is a multiple of another or the sum of multiples of other rows, then we say that the rows are linearly dependent. The rows are linearly dependent if one of them is the linear combination of any of the others. If the rows are linearly dependent, then the columns are also linearly dependent. If we interpret the rows as vectors, then we see that the second row defines the same line passing through the origin as the first. [Equation goes here - download the original to see it.] For a 2 ´ 2 matrix the only way one row can be linearly dependent on another is by that row being a multiple of the other. To illustrate the more general concept of a linear combination, for 3 ´ 3 and n ´ n matrices linear dependence can arise through one row being the sum of multiples of the other rows. Consider [Equation goes here - download the original to see it.] So P is a singular matrix. This indicates that P does not have an inverse. In fact, although none of the rows (or columns) of P is a multiple of any one other, each one is linearly dependently one the other two. The row vector [Equation goes here - download the original to see it.] That is the third row is twice the first row - the second row. We say that the third row is a linear combination of the other two rows. That would also apply to the columns. Each column is a linear combination of the other two. Let us find one such linear combination for P. Since det P = 0 we know that the first column is a linear combination of the other two. Then let [Equation goes here - download the original to see it.] Uncoupling means to unite each row of this vector equation separately. Hence [Equation goes here - download the original to see it.] These are three simultaneous equations in two unknowns. If P was not linearly dependent and having zero determinant, then this system would not have a unique solution. However, since det P = 0, we know it does. [Equation goes here - download the original to see it.] For a linearly independent (non-singular) matrix we cannot obtain a unique solution to the attempt to unite one column (or row) as a linear combination of the others. For example, [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Solving equations (1) & (2) gives [Equation goes here - download the original to see it.]but when we substitute these values into (3) we obtain [Equation goes here - download the original to see it.] which is a contradiction. Since this is a contradiction it follows that the first column cannot be a linear combination of the other two. We need to define linear dependence for vectors in general.
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Contents of
Linear dependence, independence, and singular and non-sinular matrices
1 Linear dependence, independence, and singular and non-singular matrices
2 Linear dependence
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