Does linear independence mean no solution?

If you get only the trivial solution (all coefficients zero), the vectors are linearly independent. If you get any solution other than the trivial solution, the vectors are linearly dependent.

How do you determine if a set of vectors is 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.

What does it mean for two solutions to be linearly independent?

This is a system of two equations with two unknowns. The determinant of the corresponding matrix is the Wronskian. Hence, if the Wronskian is nonzero at some t0, only the trivial solution exists. Hence they are linearly independent.

What is a non trivial solution?

A solution or example that is not trivial. Often, solutions or examples involving the number zero are considered trivial. Nonzero solutions or examples are considered nontrivial. For example, the equation x + 5y = 0 has the trivial solution (0, 0). Nontrivial solutions include (5, –1) and (–2, 0.4).

Is non trivial linearly dependent?

If there is a nontrivial combination of the vectors that adds to 0 then the vectors are called linearly dependent.

What is a non trivial linear combination?

Definition: A linear combination a1v1 + + anvn is called trivial if all the a’s are zero. Otherwise it is nontrivial. If there is a nontrivial combination of the vectors that adds to 0 then the vectors are called linearly dependent. Dependence means that there is some redundancy in the vectors.

What does a non zero determinant mean?

If a matrix’s determinant is nonzero, the matrix may have a solution. If the determinant is zero, then the matrix is not invertible and thus does not have a solution because one of the rows can be eliminated by matrix substitution of another row in the matrix.

How do you know if two solutions are linearly independent?

What is the space of linear operators of a vector space?

A covector f ∗ acts on a vector v and produces a number f ∗ (v). The space of linear maps (homomorphisms) V → W is Hom (V, W ). The space of linear operators (also called endomorphisms) of a vector space V , i.e. the space of all linear maps V → V , is End V .

Can two linearly independent functions have a zero Wronskian?

In fact, it is possible for two linearly independent functions to have a zero Wronskian! This fact is used to quickly identify linearly independent functions and functions that are liable to be linearly dependent. Example 2 Verify the fact using the functions from the previous example.

What are the norms of a vector?

Vector norms Euclidean norm: $L_2$ Manhattan norm: $L_1$ Max norm: $L_\\infty$ Vector inner product, length, and distance Vector angles and orthogonality Systems of linear equations

What is the difference between linearly dependent and linearly independent functions?

If we can find constants c1, c2, …, cn with at least two non-zero so that (2) is true for all x then we call the functions linearly dependent. If, on the other hand, the only constants that make (2) true for x are c1 = 0, c2 = 0, …, cn = 0 then we call the functions linearly independent.