Understanding Linear Independence of Vector-Valued Functions

The video tutorial explains how to determine if vector-valued functions are linearly independent or dependent. It introduces two methods: the vector equation approach and the Wronskian approach. The vector equation method involves setting up a system of equations and solving for coefficients, while the Wronskian method involves calculating the determinant of a matrix formed by the vectors. Both methods demonstrate that the given vector functions are linearly independent.

What is the condition for vector-valued functions to be linearly independent?

• The only solution to the vector equation is when all coefficients are zero.

• The functions are orthogonal to each other.

• The coefficients of the functions are all non-zero.

• The sum of the functions equals a non-zero vector.

In the first method, what is the initial step to determine linear independence?

• Check if the functions are orthogonal.

• Find the inverse of the matrix.

• Write the vector equation.

• Calculate the Wronskian determinant.

What happens when you add the two equations in the system of equations?

• The equations become dependent.

• The equations become identical.

• The second terms cancel out.

• The first terms cancel out.

Why must C2 be zero in the system of equations?

• Because e to the three T is never zero.

• Because C1 is zero.

• Because e to the three T is zero.

• Because the equations are dependent.

What is the Wronskian used for in the second method?

• To find the inverse of a matrix.

• To determine if the vector-valued functions are linearly independent.

• To solve the system of equations.

• To calculate the eigenvalues.

What does a non-zero Wronskian determinant indicate?

• The functions are linearly dependent.

• The functions are linearly independent.

• The functions are identical.

• The functions are orthogonal.

What is the determinant of the matrix in the second method?

• 8 e to the two t

• 14 e to the two t

• 6 e to the two t

• 12 e to the two t

What is the significance of the determinant being 14 e to the two t?

• It shows the functions are identical.

• It shows the functions are orthogonal.

• It shows the functions are independent.

• It shows the functions are dependent.

Which method requires the matrix to be square?

• The vector equation method.

• The Wronskian determinant method.

• The orthogonal method.

• The eigenvalue method.

What is the final conclusion about the vector-valued functions X1 and X2?

• They are orthogonal.

• They are linearly dependent.

• They are linearly independent.

• They are identical.