Transcribed Image Text:**Chapter 4: Vector Spaces**
**Exercises:**
2. **Part b.** Find specific vectors \( \mathbf{u} \) and \( \mathbf{v} \) in \( W \) such that \( \mathbf{u} + \mathbf{v} \) is not in \( W \). This is enough to show that \( W \) is not a vector space.
3. **Problem 3.** Let \( H \) be the set of points inside and on the unit circle in the \( xy \)-plane. That is, let \( H = \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : x^2 + y^2 \leq 1 \right\} \). Find a specific example—two vectors or a vector and a scalar—to show that \( H \) is not a subspace of \( \mathbb{R}^2 \).
4. Construct a geometric figure that illustrates why a line in \( \mathbb{R}^2 \) not through the origin is not closed under vector addition.
**In Exercises 5–8**, determine if the given set is a subspace of \( P_n \) for an appropriate value of \( n \). Justify your answers.
5. All polynomials of the form \( p(t) = at^2 \), where \( a \) is in \( \mathbb{R} \).
6. All polynomials of the form \( p(t) = a + t^2 \), where \( a \) is in \( \mathbb{R} \).
7. All polynomials of degree at most 3, with integers as coefficients.
8. All polynomials in \( P_n \) such that \( p(0) = 0 \).
9. Let \( H \) be the set of all vectors of the form \( \begin{pmatrix} s \\ 3s \\ 2s \end{pmatrix} \). Find a vector \( \mathbf{v} \) in \( \mathbb{R}^3 \) such that \( H = \text{Span}\{ \mathbf{v} \} \). Why does this show that \( H \) is a subspace of \( \mathbb{R}^3 \)?
10. Let \( H \) be the set of all vectors of
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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