Consider the vectors ${\mathbf{x}}_1={(8,6)}^T$ and ${\mathbf{x}}_2={(4,-1)}^T$ in ${ℝ}^2$.

1. Determine the length of each vector.

2. Let ${\mathbf{x}}_3={\mathbf{x}}_1+{\mathbf{x}}_2$. Determine the length of ${\mathbf{x}}_3$. How does its length compare with the sum of the lengths of ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$?

3. Draw a graph illustrating how ${\mathbf{x}}_3$ can be constructed geometrically using ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$. Use this graph to give a geometrical interpretation of your answer to the question in part (b).

Repeat Exercise 1 for the vectors ${\mathbf{x}}_1=(2,1{)}^T$ and ${\mathbf{x}}_2=(6,3{)}^T$.

Let C be the set of complex numbers. Define addition on C by

and define scalar multiplication by

for all real numbers $\alpha$. Show that C is a vector space with these operations.

Show that ${ℝ}^{m\times n}$, together with the usual addition and scalar multiplication of matrices, satisfies the eight axioms of a vector space.

Show that $C[a,b]$, together with the usual scalar multiplication and addition of functions, satisfies the eight axioms of a vector space.

Let P be the set of all polynomials. Show that P, together with the usual addition and scalar multiplication of functions, forms a vector space.

Show that the element 0 in a vector space is unique.

Let x, y, and z be vectors in a vector space V. Prove that if

then $\mathbf{y}=\mathbf{z}$.

Let V be a vector space and let $\mathbf{x}\in V$. Show that

1. $\beta \mathbf{0}=\mathbf{0}$ for each scalar $\beta$.

2. if $\alpha \mathbf{x}=\mathbf{0}$, then either $\alpha =0$ or $\text{x}=\text{0}$.

Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and addition on S by

We use the symbol ⊕ to denote the addition operation for this system in order to avoid confusion with the usual addition $\mathbf{x}+\mathbf{y}$ of row vectors. Show that S, together with the ordinary scalar multiplication and the addition operation ⊕, is not a vector space. Which of the eight axioms fail to hold?

Let V be the set of all ordered pairs of real numbers with addition defined by

and scalar multiplication defined by

Scalar multiplication for this system is defined in an unusual way, and consequently, we use the symbol ◦ to avoid confusion with the ordinary scalar multiplication of row vectors. Is V a vector space with these operations? Justify your answer.

Let $R^{+}$ denote the set of positive real numbers. Define the operation of scalar multiplication, denoted ◦, by

for each $x\in R^{+}$ and for any real number $\alpha$. Define the operation of addition, denoted ⊕, by

Thus, for this system, the scalar product of −3 times $\frac{1}{2}$ is given by

and the sum of 2 and 5 is given by

Is $R^{+}$ a vector space with these operations? Prove your answer.

Let R denote the set of real numbers. Define scalar multiplication by

and define addition, denoted ⊕, by

Is R a vector space with these operations? Prove your answer.

Let Z denote the set of all integers with addition defined in the usual way and define scalar multiplication, denoted ◦, by

where [[$\left[\right[\alpha \left]\right]$ denotes the greatest integer less than or equal to $\alpha$. For example,

Show that Z, together with these operations, is not a vector space. Which axioms fail to hold?

Let S denote the set of all infinite sequences of real numbers with scalar multiplication and addition defined by

Show that S is a vector space.

We can define a one-to-one correspondence between the elements of $P_n$ and ${ℝ}^n$ by

Show that if $p\leftrightarrow \mathbf{a}$ and $q\leftrightarrow \mathbf{b}$, then

1. $\alpha p\leftrightarrow \alpha \mathbf{a}$ for any scalar $\alpha$.

2. $p+q\leftrightarrow \mathbf{a}+\mathbf{b}$.

[In general, two vector spaces are said to be isomorphic if their elements can be put into a one-to-one correspondence that is preserved under scalar multiplication and addition as in (a) and (b).]