Problem 4 Let V be the span of the following vectors F 3 V1 = U2 = -2 2 V3 = 8 6 U4 = V5 = V6 = -1 5 1. Why these six vectors are NOT linearly independent? Explain? 2. Find a basis for V = span{v1, U2, U3, U4: U5, U6). 3. What is the dimension of V? Why? The following vector is in V, write it as a linear combination of the basis vectors you found in part (2). u= [2 2 2 10]"
Problem 4 Let V be the span of the following vectors F 3 V1 = U2 = -2 2 V3 = 8 6 U4 = V5 = V6 = -1 5 1. Why these six vectors are NOT linearly independent? Explain? 2. Find a basis for V = span{v1, U2, U3, U4: U5, U6). 3. What is the dimension of V? Why? The following vector is in V, write it as a linear combination of the basis vectors you found in part (2). u= [2 2 2 10]"
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Base of a vector space
![### Problem 4
**Let \( V \) be the span of the following vectors:**
\[ v_1 = \begin{bmatrix} -1 \\ 1 \\ 3 \\ 5 \end{bmatrix}, v_2 = \begin{bmatrix} -2 \\ 1 \\ 2 \\ -4 \end{bmatrix}, v_3 = \begin{bmatrix} -4 \\ -1 \\ 8 \\ 6 \end{bmatrix}, v_4 = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 1 \end{bmatrix}, v_5 = \begin{bmatrix} -2 \\ 0 \\ 6 \\ 2 \end{bmatrix}, v_6 = \begin{bmatrix} 1 \\ -1 \\ 5 \\ 7 \end{bmatrix} \]
1. **Why these six vectors are NOT linearly independent? Explain?**
2. **Find a basis for \( V = \text{span} \{ v_1, v_2, v_3, v_4, v_5, v_6 \} \).**
3. **What is the dimension of \( V \)? Why? The following vector is in \( V \), write it as a linear combination of the basis vectors you found in part (2).**
\[ u = \begin{bmatrix} 2 \\ -2 \\ 2 \\ 10 \end{bmatrix}^T \]
**Solution 4:**
(Note: This problem set includes a series of vectors and questions related to linear independence, finding a basis, and dimensionality within the context of vector spaces.)
### Explanation of Linear Independence
Linear independence of vectors means that no vector in the set can be written as a linear combination of the others. If the vectors are not linearly independent, it means there exists a non-trivial linear combination of these vectors that equals the zero vector.
### Visual Explanation of Basis and Dimension
Finding a basis for a vector space generally involves determining the set of linearly independent vectors that span the entire vector space. The dimension of the space is the number of vectors in the basis.
### Detailed Analysis Steps for Solutions
- **Step 1: Check Linear Independence**
- Construct a matrix with vectors \( v_1, v_2, v_3, v_4, v_5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe52e616-7bd7-4d85-a476-7e5fcec53ee5%2Fbe0042f1-604c-4700-8f40-26cde41056e5%2Fxgdmre6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 4
**Let \( V \) be the span of the following vectors:**
\[ v_1 = \begin{bmatrix} -1 \\ 1 \\ 3 \\ 5 \end{bmatrix}, v_2 = \begin{bmatrix} -2 \\ 1 \\ 2 \\ -4 \end{bmatrix}, v_3 = \begin{bmatrix} -4 \\ -1 \\ 8 \\ 6 \end{bmatrix}, v_4 = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 1 \end{bmatrix}, v_5 = \begin{bmatrix} -2 \\ 0 \\ 6 \\ 2 \end{bmatrix}, v_6 = \begin{bmatrix} 1 \\ -1 \\ 5 \\ 7 \end{bmatrix} \]
1. **Why these six vectors are NOT linearly independent? Explain?**
2. **Find a basis for \( V = \text{span} \{ v_1, v_2, v_3, v_4, v_5, v_6 \} \).**
3. **What is the dimension of \( V \)? Why? The following vector is in \( V \), write it as a linear combination of the basis vectors you found in part (2).**
\[ u = \begin{bmatrix} 2 \\ -2 \\ 2 \\ 10 \end{bmatrix}^T \]
**Solution 4:**
(Note: This problem set includes a series of vectors and questions related to linear independence, finding a basis, and dimensionality within the context of vector spaces.)
### Explanation of Linear Independence
Linear independence of vectors means that no vector in the set can be written as a linear combination of the others. If the vectors are not linearly independent, it means there exists a non-trivial linear combination of these vectors that equals the zero vector.
### Visual Explanation of Basis and Dimension
Finding a basis for a vector space generally involves determining the set of linearly independent vectors that span the entire vector space. The dimension of the space is the number of vectors in the basis.
### Detailed Analysis Steps for Solutions
- **Step 1: Check Linear Independence**
- Construct a matrix with vectors \( v_1, v_2, v_3, v_4, v_5
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