We are asked to find a basis for the subspace of R4 spanned by the set of vectors:
{(−1,2,1,0),(2,0,3,−1),(4,4,11,−3),(3,−2,2,−1)}
To determine a basis, we need to check for the linear independence of these vectors. If the vectors are linearly independent, they form a basis. If not, we will find a linearly independent subset that spans the same subspace.
Steps to find a basis:
- Form the matrix with the vectors as columns:
-1 & 2 & 4 & 3 \\
2 & 0 & 4 & -2 \\
1 & 3 & 11 & 2 \\
0 & -1 & -3 & -1
\end{pmatrix}$$
2. **Perform row reduction (Gaussian elimination) to find the rank of the matrix.**
We need to transform this matrix to row echelon form (REF) and identify the pivot columns. The number of pivot columns gives us the rank of the matrix, which is the number of linearly independent vectors.
Let's perform the row reduction:
- Start with the first column and make the entries below the pivot equal to zero.
We perform Gaussian elimination steps and get the row echelon form of the matrix. After these operations, the rank will tell us how many linearly independent vectors there are. The pivot columns correspond to the linearly independent vectors.
### Row reduction steps:
1. $$R_2 \rightarrow R_2 + 2R_1$$
2. $$R_3 \rightarrow R_3 + R_1$$
3. $$R_4 \rightarrow R_4 + R_1$$
4. Continue eliminating and simplifying, until you reach REF.
Once we get the row echelon form, the non-zero rows indicate the number of linearly independent vectors. The columns with pivots will correspond to the original vectors that form the basis.
I'll calculate the row-reduced form and identify the linearly independent columns. sympy as sp
# Define the matrix
A = sp.Matrix([
[-1, 2, 4, 3],
[2, 0, 4, -2],
[1, 3, 11, 2],
[0, -1, -3, -1]
])
# Perform row reduction
RREF, pivot_columns = A.rref()
RREF, pivot_columnsThe row echelon form (RREF) of the matrix is:
$$\begin{pmatrix}
1 & 0 & 2 & -1 \\
0 & 1 & 3 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}$$
From the row-reduced matrix, we can see that the first two columns contain pivots, indicating that the first and second vectors are linearly independent. The other columns do not contain pivots, which means that the third and fourth vectors can be written as linear combinations of the first two.
### Conclusion:
The vectors corresponding to the pivot columns are:
$$(-1, 2, 1, 0) \quad \text{and} \quad (2, 0, 3, -1)$$
Therefore, a basis for the subspace spanned by the given set of vectors is:
$$\{ (-1, 2, 1, 0), (2, 0, 3, -1) \}$$
Would you like a more detailed explanation of the row reduction steps or have any further questions on this topic?
Here are some related questions to explore:
1. How do you find the rank of a matrix and its relation to linear independence?
2. What is the geometric interpretation of a basis in $$\mathbb{R}^4$$?
3. How can you test for linear dependence using the determinant of a matrix?
4. What is the difference between row echelon form and reduced row echelon form?
5. How do you perform a change of basis in $$\mathbb{R}^4$$?
**Tip**: When performing row reduction, always keep an eye on pivot positions. They directly tell you about the linear independence of the columns.