Please help solve the question shown in the photo. The topic is "Linear Algebra". Thank you!
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.
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
![**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb9a7c18-65f1-48ed-bb7e-4937a04e4157%2F91c6a703-5c2d-4860-8da6-71371aade0e0%2Fg5qfdsl_processed.png&w=3840&q=75)
