Find the basis and rank. 4s -3s:s,t E R

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please help solve the question shown in the photo. The topic is "Linear Algebra". Thank you!

**Find the basis and rank.**

Given the set:

\[
\left\{ \begin{bmatrix} 4s \\ -3s \\ -t \end{bmatrix} : s, t \in \mathbb{R} \right\}
\]

To find the basis and rank of this vector space, we express the vector as a linear combination of vectors involving \(s\) and \(t\):

\[
s \begin{bmatrix} 4 \\ -3 \\ 0 \end{bmatrix} + t \begin{bmatrix} 0 \\ 0 \\ -1 \end{bmatrix}
\]

This indicates the space is spanned by the vectors:

1. \(\begin{bmatrix} 4 \\ -3 \\ 0 \end{bmatrix}\)
2. \(\begin{bmatrix} 0 \\ 0 \\ -1 \end{bmatrix}\)

These vectors are linearly independent. Therefore, the basis of the vector space consists of these two vectors.

**Rank:**

The rank of the vector space, given by the number of vectors in the basis, is 2.
Transcribed Image Text:**Find the basis and rank.** Given the set: \[ \left\{ \begin{bmatrix} 4s \\ -3s \\ -t \end{bmatrix} : s, t \in \mathbb{R} \right\} \] To find the basis and rank of this vector space, we express the vector as a linear combination of vectors involving \(s\) and \(t\): \[ s \begin{bmatrix} 4 \\ -3 \\ 0 \end{bmatrix} + t \begin{bmatrix} 0 \\ 0 \\ -1 \end{bmatrix} \] This indicates the space is spanned by the vectors: 1. \(\begin{bmatrix} 4 \\ -3 \\ 0 \end{bmatrix}\) 2. \(\begin{bmatrix} 0 \\ 0 \\ -1 \end{bmatrix}\) These vectors are linearly independent. Therefore, the basis of the vector space consists of these two vectors. **Rank:** The rank of the vector space, given by the number of vectors in the basis, is 2.
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