Chapter 4, Vector spaces Video Solutions, Linear Algebra

Problem 1

For each of the following sets, either show that the set is a vector space or explain why it is not a vector space.
(a) The set $\mathbb{R}$ of real numbers under the usual operations of addition and multiplication.
(b) The set $\{(x, 0) \mid x \in \mathbb{R}\}$ under the usual operations of addition and multiplication on $\mathbb{R}^{2}$.
(c) The set $\{(x, 1) \mid x \in \mathbb{R}\}$ under the usual operations of addition and multiplication on $\mathbb{R}^{2}$.
(d) The set $\{(x, 0) \mid x \in \mathbb{R}, x \geq 0\}$ under the usual operations of addition and multiplication on $\mathbb{R}^{2}$.
(e) The set $\{(x, 1) \mid x \in \mathbb{R}, x \geq 0\}$ under the usual operations of addition and multiplication on $\mathbb{R}^{2}$.
(f) The set $\left\{\left[\begin{array}{cc}a & a+b \\ a+b & a\end{array}\right] \mid a, b \in \mathbb{R}\right\}$ under the usual operations of addition and multiplication on $\mathbb{R}^{2 \times 2}$.
(g) The set $\left\{\left[\begin{array}{cc}a & a+b+1 \\ a+b & a\end{array}\right] \mid a, b \in \mathbb{R}\right\}$ under the usual operations of addition and multiplication on $\mathbb{R}^{2 \times 2}$. under the usual operations of addition.

Anthony Ramos

Numerade Educator

01:37

Problem 2

Show that the space $V=\left\{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{F}^{3} \mid x_{1}+2 x_{2}+2 x_{3}=0\right\}$ forms a vector space.

Dharmendra Jain

Numerade Educator

03:52

Problem 3

For each of the following sets, either show that the set is a subspace of $\mathcal{C}(\mathbb{R})$ or explain why it is not a subspace.
(a) The set $\{f \in \mathcal{C}(\mathbb{R}) \mid f(x) \leq 0, \forall x \in \mathbb{R}\}$.
(b) The set $\{f \in \mathcal{C}(\mathbb{R}) \mid f(\mathrm{o})=\mathbf{o}\}$.
(c) The set $\{f \in \mathcal{C}(\mathbb{R}) \mid f(0)=2\}$.
(d) The set of all constant functions.
(e) The set $\{\alpha+\beta \sin (x) \mid \alpha, \beta \in \mathbb{R}\}$.

Khoobchandra Agrawal

Numerade Educator

02:44

Problem 4

Give an example of a nonempty subset $U \subset \mathbb{R}^{2}$ such that $U$ is closed under scalar multiplication but is not a subspace of $\mathbb{R}^{2}$.

Sanchit Jain

Numerade Educator

02:35

Problem 5

Let $\mathbb{F}[z]$ denote the vector space of all polynomials having coefficient over $\mathbb{F}$, and define $U$ to be the subspace of $\mathbb{F}[z]$ given by
$$
U=\left\{a z^{2}+b z^{5} \mid a, b \in \mathbb{F}\right\}
$$
Find a subspace $W$ of $\mathbb{F}[z]$ such that $\mathbb{F}[z]=U \oplus W$.

Doruk Isik

Numerade Educator