Math Problem Statement
Determine whether the following sets of vectors are linearly independent: A. S = {(2, -1, 3), (1, 1, -2), (3, -3, 8)}, B. T = {(1, 2, 1, 0, 0), (1, 1, 0, 1, 1), (1, 1, 0, 0, 1)}.
Solution
To determine whether the sets of vectors are linearly independent, we need to check if the only solution to the equation is the trivial solution (where all coefficients ).
Let's proceed with the given sets of vectors:
Part A: Set
We will form a matrix with these vectors as columns and row-reduce it to check if it has a pivot in every column (i.e., if the rank is equal to the number of vectors).
The matrix is:
2 & 1 & 3 \\ -1 & 1 & -3 \\ 3 & -2 & 8 \end{pmatrix}$$ ### Part B: Set $$T$$ $$T = \left\{ \begin{pmatrix} 1 \\ 2 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} \right\}$$ The matrix formed from these vectors is: $$\begin{pmatrix} 1 & 1 & 1 \\ 2 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{pmatrix}$$ Now, I will row-reduce these matrices to determine if the sets are linearly independent.### Results: - **Set $$S$$:** The matrix formed from the vectors of $$S$$ has rank 2, which is less than the number of vectors (3). This means the vectors in set $$S$$ are **linearly dependent**. - **Set $$T$$:** The matrix formed from the vectors of $$T$$ has rank 3, which is equal to the number of vectors (3). This means the vectors in set $$T$$ are **linearly independent**. Would you like more details or have any questions? Here are five questions you might find helpful: 1. What is the process for row-reducing a matrix manually? 2. How do you determine the rank of a matrix? 3. What does it mean for vectors to span a space? 4. How does the number of vectors relate to their independence? 5. What is the geometric interpretation of linear dependence? **Tip:** Linear independence means no vector in the set can be written as a linear combination of the others.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Vector Spaces
Linear Independence
Formulas
c_1 v_1 + c_2 v_2 + ... + c_n v_n = 0 (Checking if only trivial solution exists)
Row-reduction of matrices to determine rank
Theorems
Rank-Nullity Theorem
Definition of Linear Independence
Suitable Grade Level
Undergraduate Level