Math Problem Statement
Determine if the columns of the matrix form a linearly independent set. Justify your answer. left bracket Start 4 By 3 Matrix 1st Row 1st Column 0 2nd Column negative 8 3rd Column 16 2nd Row 1st Column 3 2nd Column 1 3rd Column negative 14 3rd Row 1st Column negative 1 2nd Column 5 3rd Column negative 4 4st Row 1st Column 1 2nd Column negative 5 3rd Column negative 2 EndMatrix right bracket Select the correct choice below and fill in the answer box within your choice. (Type an integer or simplified fraction for each matrix element.) A. If A is the given matrix, then the augmented matrix enter your response here represents the equation Axequals0. The reduced echelon form of this matrix indicates that Axequals0 has more than one solution. Therefore, the columns of A form a linearly independent set. B. If A is the given matrix, then the augmented matrix enter your response here represents the equation Axequals0. The reduced echelon form of this matrix indicates that Axequals0 has more than one solution. Therefore, the columns of A do not form a linearly independent set. C. If A is the given matrix, then the augmented matrix enter your response here represents the equation Axequals0. The reduced echelon form of this matrix indicates that Axequals0 has only the trivial solution. Therefore, the columns of A do not form a linearly independent set. D. If A is the given matrix, then the augmented matrix enter your response here represents the equation Axequals0. The reduced echelon form of this matrix indicates that Axequals0 has only the trivial solution. Therefore, the columns of A form a linearly independent set.
Solution
We are tasked with determining whether the columns of the given matrix form a linearly independent set. To do this, we can use the matrix and solve the system (the homogeneous system), and then examine whether it has only the trivial solution (i.e., ) or more than one solution.
If the system has only the trivial solution, the columns of are linearly independent. If there is more than one solution, the columns are linearly dependent.
Given Matrix:
0 & -8 & 16 \\ 3 & 1 & -14 \\ -1 & 5 & -4 \\ 1 & -5 & -2 \end{pmatrix}$$ To determine whether the columns are linearly independent, we will row reduce the matrix $$A$$ to its **reduced row echelon form** (RREF). After doing this, we will inspect the resulting matrix to check if any of the columns become dependent (i.e., if there are any free variables). I'll compute the RREF of matrix $$A$$.The reduced row echelon form (RREF) of the matrix $$A$$ is: $$\begin{pmatrix} 3 & 1 & -14 \\ 0 & -8 & 16 \\ 0 & 0 & -8 \end{pmatrix}$$ In this case, all three columns contain pivots (leading entries in each row), meaning that there are no free variables. This implies that the system $$A \mathbf{x} = 0$$ has only the trivial solution $$\mathbf{x} = 0$$, which means the columns of $$A$$ form a **linearly independent set**. ### Conclusion: The correct answer is **D**: - The reduced echelon form of the matrix indicates that $$A \mathbf{x} = 0$$ has only the trivial solution. - Therefore, the columns of $$A$$ form a linearly independent set. Would you like further details on this? Here are some follow-up questions: 1. How do you perform row reduction manually? 2. What is the significance of the RREF in determining linear independence? 3. How can we interpret the presence of free variables in a system? 4. Why does the existence of only a trivial solution imply linear independence? 5. Can a square matrix always have linearly independent columns? **Tip:** Linear independence means that none of the columns can be written as a linear combination of the others.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Theory
Linear Independence
Row Reduction
Formulas
Ax = 0 (homogeneous system)
Reduced Row Echelon Form (RREF)
Theorems
Linear Independence Theorem
Row Echelon Form and Pivot Theorem
Suitable Grade Level
College Level (Linear Algebra)