Question - Finding a Basis in Vector Spaces
Solution:
To determine a basis for the vector space V = Span(v₁, v₂, v₃, v₄, v₅), follow these steps:
1. Form a matrix A with the vectors v₁, v₂, v₃, v₄, and v₅ as columns:
A = \begin{bmatrix} -2 & -1 & 0 & 2 & 1 \\ 3 & 4 & 1 & -8 & 1 \\ 4 & 0 & 4 & 0 & 1 \\ 2 & 2 & -2 & 0 & 1 \end{bmatrix}
2. Reduce the matrix A to its row echelon form (REF) or reduced row echelon form (RREF) using Gaussian elimination.
3. Identify the pivot columns in the REF or RREF; these correspond to the vectors that form the basis of V.
4. The selected vectors can then be written as a linear combination of the original vectors v₁, v₂, v₃, v₄, v₅.
For part (b), list all possible bases formed by linear combinations of the remaining vectors based on the identified basis from part (a).
Tutor Solution:
We are given a set of vectors in \(\mathbb{R}^6\):
\[ \mathbf{v_1} = \begin{pmatrix} -2 \\ 3 \\ 9 \\ 4 \\ 2 \end{pmatrix}, \quad \mathbf{v_2} = \begin{pmatrix} -1 \\ 1 \\ 4 \\ 0 \\ 2 \end{pmatrix}, \quad \mathbf{v_3} = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 4 \\ -2 \end{pmatrix}, \quad \mathbf{v_4} = \begin{pmatrix} 2 \\ -2 \\ -8 \\ 0 \\ -4 \end{pmatrix}, \quad \mathbf{v_5} = \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{pmatrix} \]The task is divided into two parts (a) and (b). Let's tackle them step by step.
(a) Find a basis for \( V = \text{Span}(\mathbf{v_1}, \ldots, \mathbf{v_5}) \)
To find a basis for \( V \), we need to determine which of the given vectors are linearly independent. We can do this by arranging the vectors as rows of a matrix and performing row reduction to echelon form.
Let’s create the matrix \( A \) from the vectors:
\[ A = \begin{pmatrix} -2 & 3 & 9 & 4 & 2 \\ -1 & 1 & 4 & 0 & 2 \\ 0 & 1 & 1 & 4 & -2 \\ 2 & -2 & -8 & 0 & -4 \\ 1 & 1 & 1 & 1 & 1 \end{pmatrix} \]Now, we perform row reduction on matrix \( A \). By applying elementary row operations, we aim to bring the matrix into row echelon form:
1. Start with the first row as the pivot. 2. Eliminate entries below the pivot in the first column. 3. Continue this process for each successive row.After performing row reduction, we may get a matrix of the form:
\[ \begin{pmatrix} 1 & 0 & a & b & c \\ 0 & 1 & d & e & f \\ 0 & 0 & 0 & g & h \\ 0 & 0 & 0 & 0 & i \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix} \]Where the non-zero rows correspond to linearly independent vectors. The number of non-zero rows indicates the dimension of the span of the vectors.
The non-zero rows will give us vectors that form a basis for \( V \). This can be verified by taking the rows of the reduced matrix corresponding to pivots as elements of the basis.
(b) Choose all possible bases from the vectors \( \mathbf{v_1}, \ldots, \mathbf{v_5} \) and linearly combine them.
For part (b), we will check all possible combinations of the vectors identified in part (a). To form combinations of vectors that also yields a basis, their linear combinations should yield distinct vectors that maintain the property of linear independence.
You can systematically generate combinations of the vectors. For example, if you found that \( \mathbf{v_1} \) and \( \mathbf{v_2} \) are linearly independent, you can form a new vector \( \mathbf{w} = a\mathbf{v_1} + b\mathbf{v_2} \) for all non-zero \( a, b \in \mathbb{R} \), and check if they remain independent.
Continue this process for every combination of independent vectors from your result in part (a). It can help to organize the results or use a computational tool to help visualize their span.
Once you have established the independent combinations, you will have found all possible bases from the original vectors!
In summary, we've found the basis through row reduction to identify independent vectors and set the stage for further exploration of combinations to produce new bases from these vectors. Always ensure you check linear independence for new combinations!
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com