Can a matrix with a zero row be linearly independent?

The system of rows is called linearly independent, if only trivial linear combination of rows are equal to the zero row (there is no non-trivial linear combination of rows equal to the zero row).

What does it mean when the last row of a matrix is all zeros?

Row-Echelon Form If there is a row of all zeros, then it is at the bottom of the matrix. The first non-zero element of any row is a one. That element is called the leading one. The leading one of any row is to the right of the leading one of the previous row.

How do you tell if the columns of a matrix are linearly independent?

Given a set of vectors, you can determine if they are linearly independent by writing the vectors as the columns of the matrix A, and solving Ax = 0. If there are any non-zero solutions, then the vectors are linearly dependent. If the only solution is x = 0, then they are linearly independent.

How do you know if a column vector is linearly dependent?

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.

How many solutions does a 3×3 matrix have?

A 3×3 matrix equation Ax=b is solved for two different values of b. In one case there is no solution, and in another there are infinitely many solutions.

Can 3 linearly dependent vectors span R3?

Yes. The three vectors are linearly independent, so they span R3.

How can a matrix have infinite solutions?

Note: To know about the infinite solution of a matrix first we have to check nonzero rows in the matrix. That means if the number of variables is more than nonzero rows then that matrix has an infinite solution.

Is the last row of a matrix always zeros?

The answer is no whether or not the last row is zeros. The last row can be any linear combination of the first two rows leading to infinitely many solutions of the linear system, and the column vectors will always be linearly dependent.

Can column vectors be linearly independent in a 3×3 matrix?

If a 3×3 matrix has infinitely many solutions (last row is zeros), can the column vectors be linearly independent? Introducing MongoDB 5.0. Native time series platform on MongoDB 5.0 makes it faster and easier to build IoT and financial apps. The answer is no whether or not the last row is zeros.

How many solutions can a matrix have if the value is 0?

If a = 0, the equation gives us no new information and we really just have one useful equation, given by the second row; depending on the rest of the matrix, there could still be zero, one, or infinitely many solutions to the system of equations. Additionally, consider multiplication by a matrix A whose i th row consists of all 0s.

Is the last row of a linear system always zeros?

, PhD in Applied Mathematics. The answer is no whether or not the last row is zeros. The last row can be any linear combination of the first two rows leading to infinitely many solutions of the linear system, and the column vectors will always be linearly dependent.