thrive said:it would mean there is some linear combination of c1, c2, c3 (not all zero) that would solve that first equation.
In the second equation it would imply that not necessarily would there be a c4 to cancel the V4 term?
Linear independence refers to a set of vectors in a vector space that cannot be written as a linear combination of each other. In other words, none of the vectors can be expressed as a scalar multiple of another vector in the set.
To determine if V1-V4 in R4 are linearly independent, you can use the row reduction method to create an augmented matrix and then check if the matrix has a unique solution. If there is a unique solution, then the vectors are linearly independent. If the matrix has no solution or infinitely many solutions, then the vectors are linearly dependent.
Yes, a set of vectors can be linearly independent in one vector space but dependent in another. This is because the definition of linear independence is based on the vector space itself, and different vector spaces can have different dimensions and bases.
No, the statement "V1-V4 in R4 are linearly independent" is not always true. It depends on the specific values of V1-V4 and the dimension of R4. For example, if V1 = [1, 0, 0, 0], V2 = [0, 1, 0, 0], V3 = [0, 0, 1, 0], and V4 = [0, 0, 0, 1], then the statement would be true. However, if any of the vectors were a linear combination of the others, the statement would be false.
Linear independence is closely related to the concept of a basis. A basis for a vector space is a set of linearly independent vectors that span the entire space. This means that any vector in the space can be written as a linear combination of the basis vectors. In other words, the basis vectors form a coordinate system for the vector space. If a set of vectors is linearly dependent, it cannot form a basis for the space.