Math Problem Statement
The image contains some mathematical content involving equations and a matrix problem. Let’s analyze and solve the problem step-by-step.
Solution
The image contains some mathematical content involving equations and a matrix problem. Let’s analyze and solve the problem step-by-step.
Observations from the Image:
- There are two given vectors and some operations involving those vectors.
- There is a question that seems to involve verifying whether the vectors form a linearly independent set, or possibly another property involving a matrix.
Solution Steps:
Step 1: Vector Definitions and Basic Operations Let’s start by defining the vectors and any relevant equations from the image.
If the vectors are:
\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \quad \vec{v_2} = \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}, \quad \vec{v_3} = \begin{bmatrix} 7 \\ 8 \\ 9 \end{bmatrix}$$ **Step 2: Linear Independence Check** To check if these vectors are linearly independent, we set up a matrix $$A$$ with these vectors as columns and determine if the determinant of $$A$$ is zero. Matrix $$A$$: $$A = \begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{bmatrix}$$ **Step 3: Calculate the Determinant of $$A$$** $$\text{det}(A) = 1(5 \cdot 9 - 6 \cdot 8) - 4(2 \cdot 9 - 3 \cdot 8) + 7(2 \cdot 6 - 3 \cdot 5)$$ Simplify each term: $$\text{det}(A) = 1(45 - 48) - 4(18 - 24) + 7(12 - 15)$$ $$= 1(-3) - 4(-6) + 7(-3)$$ $$= -3 + 24 - 21 = 0$$ **Conclusion**: The determinant is zero, which means the vectors are linearly dependent. Would you like me to go into more details, or do you have specific questions about this process? ### Further Related Questions: 1. How do you determine if a set of vectors is linearly independent? 2. What does it mean when the determinant of a matrix is zero? 3. How can you find the rank of a matrix using its row echelon form? 4. What is the geometric interpretation of linearly dependent vectors? 5. How can you determine if vectors span a given vector space? **Tip**: For quick determinant checks, remember that a zero determinant always implies linear dependence among the columns or rows of the matrix!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrices
Determinants
Formulas
Determinant of a matrix
Matrix representation of vectors
Theorems
Properties of determinants
Linear independence criterion
Suitable Grade Level
University level (Linear Algebra)