https://file.aiquickdraw.com/mathbot/file/17262170367275soxpy08.jpg

To determine the value(s) of \( a \) that make the set of vectors \( S \) linearly dependent, we need to analyze the matrix formed by placing these vectors as columns in a matrix. The set \( S \) consists of the following vectors:

\[
S = \left\{
\begin{bmatrix} 1 \\ 2 \\ 3 \\ a \end{bmatrix}, 
\begin{bmatrix} a \\ 0 \\ -1 \\ 2 \end{bmatrix}, 
\begin{bmatrix} 0 \\ 0 \\ a^2 \\ 7 \end{bmatrix}, 
\begin{bmatrix} 1 \\ a \\ 1 \\ 1 \end{bmatrix}, 
\begin{bmatrix} 2 \\ -2 \\ 3 \\ a^3 \end{bmatrix}
\right\}
\]

These vectors will be linearly dependent if the determinant of the matrix formed by these vectors is zero. Since the number of vectors exceeds the dimension of the vector space (there are 5 vectors in a 4-dimensional space), these vectors are automatically dependent. However, we want to find specific values of \( a \) where the dependency is more significant, potentially reducing the rank of the matrix.

We will form the matrix with these column vectors and analyze its rank and determinant to find the values of \( a \). I will begin by constructing the matrix and solving for the specific conditions.Since the matrix is not square (it's a 4x5 matrix), calculating the determinant directly is not possible. However, we can still analyze the linear dependence of these vectors by determining when the rank of the matrix is less than 4. To do this, we can set up the matrix and perform row reduction or check for rank deficiency under certain conditions for \( a \).

I'll proceed with finding the rank of the matrix and analyzing the conditions under which the rank is less than 4.It seems like I can't do more advanced data analysis right now. Please try again later.

Would you like me to walk you through the manual steps to solve this problem? If so, I can guide you through row reduction or provide more context on matrix rank.

Let me know if you have any questions, or if you'd like to explore related topics such as:

1. How to check if a matrix is linearly dependent.
2. How to compute the rank of a matrix.
3. Conditions for vector linear dependence.
4. What it means for a set of vectors to be linearly dependent.
5. How determinants relate to linear dependence in square matrices.

**Tip**: Linear dependence often occurs when one vector can be written as a combination of the others!

For what value(s) of a is the set of vectors S linearly dependent?

This problem involves determining the value of 'a' that makes the set of vectors S linearly dependent. The key concepts include linear dependence, matrix rank, and vector analysis in linear algebra. Suitable for university-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation