Table of Contents
- 1 Do row operations preserve the linear dependence relations?
- 2 Do row operations affect linear dependence relations among the columns of a matrix?
- 3 Why do row operations preserve rank?
- 4 How can you tell if something is linearly dependent?
- 5 Does row replacement change eigenvalues?
- 6 How do you find linearly independent vectors?
Do row operations preserve the linear dependence relations?
Remark: Row operations do not preserve the linear dependence relations among the rows (although they do for the columns). This means, for example, that while the first k rows of B may be linearly independent, while the first k rows of A may not be.
Do row operations affect linear dependence relations among the columns of a matrix?
Because row operations do not affect the pivot columns of a matrix, they also cannot affect linear dependence relations among the columns of a matrix OD. If a series of row operations is performed on a matrix A to form B, then linearly independent columns of A correspond to linearly independent columns of B.
Are rows in row echelon form linearly independent?
Because the row echelon form has a ”leading 1” in each column, the columns of the original matrix are linear independent.
Do row operations preserve the column space?
Elementary row operations affect the column space. So, generally, a matrix and its echelon form have different column spaces. However, since the row operations preserve the linear relations between columns, the columns of an echelon form and the original columns obey the same relations.
Why do row operations preserve rank?
An elementary row operation multiplies a matrix by an elementary matrix on the left. Those elementary matrices are invertible, so the row op- erations preserve rank. In other words, the dimension of the column space equals the dimension of the row space, and both equal the rank of the matrix.
How can you tell if something is linearly dependent?
Two vectors are linearly dependent if and only if they are collinear, i.e., one is a scalar multiple of the other. Any set containing the zero vector is linearly dependent. If a subset of { v 1 , v 2 ,…, v k } is linearly dependent, then { v 1 , v 2 ,…, v k } is linearly dependent as well.
How do you know if a matrix is dependent?
If a consistent system has exactly one solution, it is independent .
- If a consistent system has an infinite number of solutions, it is dependent . When you graph the equations, both equations represent the same line.
- If a system has no solution, it is said to be inconsistent .
Do row operations change null space?
3. Elementary row operations do not change the null space of a matrix.
Does row replacement change eigenvalues?
A row replacement operation on A does not change the eigenvalues.
How do you find linearly independent vectors?
We have now found a test for determining whether a given set of vectors is linearly independent: A set of n vectors of length n is linearly independent if the matrix with these vectors as columns has a non-zero determinant. The set is of course dependent if the determinant is zero.