HallsofIvy said:Yes, it does. What you have shown is that those three vectors are independent and independent vectors form a basis for their span.
Dick said:It would mean that. Except I think you did the row reduction wrong. Those vectors aren't linearly independent.
lolimcool said:oops i did i did it again and got
1 0 5
0 1 -3
0 0 0
so dimension = 2?
The dimension of the linear span of a vector is the minimum number of vectors needed to span the entire vector space. It represents the number of linearly independent vectors that make up the span.
The dimension of the linear span of a vector can be calculated by finding the rank of the matrix formed by the vectors, or by counting the number of non-zero rows in the reduced row echelon form of the matrix.
Yes, the dimension of the linear span of a vector can be greater than the number of vectors in the set. This is because the dimension also takes into account linear combinations of the vectors, not just the individual vectors themselves.
The dimension of the linear span of a vector is closely related to linear independence. If the dimension is equal to the number of vectors in the set, then the vectors are linearly independent. However, if the dimension is less than the number of vectors, then the vectors are linearly dependent.
Understanding the dimension of the linear span of a vector is important in linear algebra because it helps determine the size and structure of a vector space. It also allows for the identification of linearly independent vectors, which are crucial in solving systems of linear equations and other applications in mathematics and science.