Number 11 and 12 linear algebra pratice exercise please ## Vector Spaces and Subspaces### 6. Polynomials in Vector Spaces- All polynomials of the form $ p(t) = a + t^2 $, where $ a $ is in $ \mathbb{R} $.### 7. Involving Degree Constraints- All polynomials of degree at most 3, with integers as coefficients.### 8. Specific Constraints on Polynomials- All polynomials in $ \mathbb{P}_n $ such that $ p(0) = 0 $.### 9. Subspace in $ \mathbb{R}^3 $- **Problem**: Let $ H $ be the set of all vectors of the form $ \begin{bmatrix}s \\3s \\2s\end{bmatrix} $. - **Task**: Find a vector $ v $ in $ \mathbb{R}^3 $ such that $ H = \text{Span}\{v\} $. Explain why $ H $ is a subspace of $ \mathbb{R}^3 $.### 10. Another Subspace Example in $ \mathbb{R}^3 $- **Problem**: Let $ H $ be the set of all vectors of the form $ \begin{bmatrix}2t \\0 \\-t\end{bmatrix} $. - **Task**: Show that $ H $ is a subspace of $ \mathbb{R}^3 $. (Use the method of Exercise 9.)### 11. Subspace Involving Arbitrary Constants- **Problem**: Let $ W $ be the set of all vectors of the form $ \begin{bmatrix}5b + 2c \\b \\c \\0\end{bmatrix} $, where $ b $ and $ c $ are arbitrary. - **Task**: Find vectors $ u $ and $ v $ such that $ W = \text{Span}\{u, v\} $. Explain why $ W $ is a subspace of $ \mathbb{R}^4 $.### 12. Structuring a Subspace in $ \mathbb{R}^4 $- **Problem**: Let \( W

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Description: Number 11 and 12 linear algebra pratice exercise please ## Vector Spaces and Subspaces### 6. Polynomials in Vector Spaces- All polynomials of the form $ p(t) = a + t^2 $, where $ a $ is in $ \mathbb{R} $.### 7. Involving Degree Constraints- Al

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Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Number 11 and 12 linear algebra pratice exercise please

## Vector Spaces and Subspaces

### 6. Polynomials in Vector Spaces
- All polynomials of the form $ p(t) = a + t^2 $, where $ a $ is in $ \mathbb{R} $.

### 7. Involving Degree Constraints
- All polynomials of degree at most 3, with integers as coefficients.

### 8. Specific Constraints on Polynomials
- All polynomials in $ \mathbb{P}_n $ such that $ p(0) = 0 $.

### 9. Subspace in $ \mathbb{R}^3 $
- **Problem**: Let $ H $ be the set of all vectors of the form $ \begin{bmatrix}
s \\
3s \\
2s
\end{bmatrix} $.
- **Task**: Find a vector $ v $ in $ \mathbb{R}^3 $ such that $ H = \text{Span}\{v\} $. Explain why $ H $ is a subspace of $ \mathbb{R}^3 $.

### 10. Another Subspace Example in $ \mathbb{R}^3 $
- **Problem**: Let $ H $ be the set of all vectors of the form $ \begin{bmatrix}
2t \\
0 \\
-t
\end{bmatrix} $.
- **Task**: Show that $ H $ is a subspace of $ \mathbb{R}^3 $. (Use the method of Exercise 9.)

### 11. Subspace Involving Arbitrary Constants
- **Problem**: Let $ W $ be the set of all vectors of the form $ \begin{bmatrix}
5b + 2c \\
b \\
c \\
0
\end{bmatrix} $, where $ b $ and $ c $ are arbitrary.
- **Task**: Find vectors $ u $ and $ v $ such that $ W = \text{Span}\{u, v\} $. Explain why $ W $ is a subspace of $ \mathbb{R}^4 $.

### 12. Structuring a Subspace in $ \mathbb{R}^4 $
- **Problem**: Let \( W

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.

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