To find a basis of the span of the given subset X and extend it to a basis of R4, we can follow these steps: 1. Span of X: The span of X is the set of all possible linear combinations of the vectors in X. To determine the span, we need to find the values of the coefficients (scalars) that make any vector in R4 attainable through a linear combination of the vectors in X. 2. Row-echelon form: We can create an augmented matrix using the vectors in X and perform row operations to reduce it to row-echelon form. This process helps us find the values for the coefficients that satisfy the linear combination. 3. Basis for Span X: The basis for the span of X consists of the vectors from X that correspond to the pivot columns in the row-echelon form. In other words, the basis vectors are the linearly independent vectors in X. 4. Extension to a basis of R4: To extend the basis for the span of X to a basis of R4, we need to add vectors that are linearly independent of the existing basis vectors. One way to achieve this is by adding the standard basis vectors for R4, which are (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1), to the existing basis for the span of X. 5. Final basis for R4: A basis for R4 can be formed by combining the basis vectors for the span of X and the standard basis vectors for R4. Therefore, a basis for R4 can be {v1, v2, v3, v4, (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}, where v1, v2, v3, and v4 are the vectors given in X. In summary: - Basis for the span of X: {v1, v2, v3, v4} - Basis for R4: {v1, v2, v3, v4, (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)} These steps help us find a basis for the span of X and extend it to a basis of R4