Correct the following definition.

Definition (Basis): A basis for a vector space V over a field F is a set of vectors in V that span V and are linearly dependent.

Solution:
Correct Definition (Basis): A basis for a vector space V over a field F is a set of vectors in V that span V and are linearly independent.
Comment on Error: The set must be linearly independent. Linear dependence means the set is redundant. A basis must be the most efficient , non-redundant set that spans the space. END

Exercise (Basis): Does the set `B = {(1 , 0 , 0) , (0 , 1 , 0) , (1 , 1 , 0)}` form a basis for the vector space `RR^3` ?

Reformulate the Given Exercise: Check if the set B simultaneously satisfies the two conditions required for a basis of `RR^3`:
1. Does B span `RR^3` ?
2. Is B linearly independant ?

Give a general Strategy for a Solution:
1. Since `dim(RR^3)` and `abs B = 3` , we only need to check one of the two conditions (spanning or linear independence).
2. We will check for linear independence by forming the matrix M with the vectors as columns and calculating its determinant.
3. If `det(M) ne 0` , the set is linearly independent and therefore a basis. If `det(M) = 0` , the set is linearly dependent and is not a basis.

Write down the specific Theory used : A basis for a vector space V is a set of vectors in V that span V and are linearly independent.

Do the Exercise:
1. Form a Matrix: `M = [[1 , 0 , 1] , [0 , 1 , 1] , [0 , 0 , 0]]`.

2. Calculate the Determinant: We can compute the determinant by expanding along the first row or simply by recognizing the matrix is an upper triangular matrix with a zero on the main diagonal:
`det(M) = 1 * (1 * 0 - 1 * 0) - 0 * (0 * 0 - 1 * 0) + 1 * (0 * 0 - 1 * 0) = 0`.
Alternatively , the determinant of a matrix with a row or column of zeros is 0.

Final Result: Since `det(M) = 0` , the column vectors (the vectors in B) are linearly dependent. Because the set is linearly dependent , it cannot be a basis for `RR^3`. (Also , because the vectors lie in the xy-plane , they cannot span all of `RR^3`).

Pedagogical Summary (The "Perfect Fit" Theorem): The Basis Theorem acts like "Perfect Fit" rule for finite-dimensional vector spaces like `RR^n`:

Case 1: If the number of vectors in a set `B subset RR^n` is not equal to the dimension of the space n , then you must check both the spanning and the linear independence conditions to determine if B is a basis.

Case 2: When the number of vectors equals the dimension , the set is the right size. If it is linearly independent , it has to be big enough to span. If it spans , it has to be small enough to be independent. Meaning you only have to check one condition in this case.

Special Case 3: "The Lack of Space Theorem" in `RR^n`: The phrase perfectly captures the essence of The Basis Theorem , particularly the case of having too many vectors: If a set of vectors has more vectors than the dimension of the space , it must be linearly dependent. END