Problem Suppose $v_1, v_2, v_3, v_4$ is linearly independent in $V$. Prove that the list $$v_1 - v_2, v_2 - v_3, v_3 - v_4, v_4$$ is also linearly independent. Solution If $a_1, a_2, a_3, a_4 \in \mathbb{F}$, then $$0 = a_1(v_1 - v_2) + a_2(v_2 - v_3) + a_3(v_3 - v_4) + a_4v_4$$ $$a_1v_1 + (a_2 - a_1)v_2 + (a_3 - a_2)v_3 + (a_4 - a_3)v_4$$ Implies $a_1 = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = 0$ by the linear independence of $v_1, v_2, v_3, v_4$, but $a_1 = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = 0$ implies $a_1 = a_2 = a_3 = a_4 = 0$, thus the list $v_1 - v_2, v_2 - v_3, v_3 - v_4, v_4$ is linearly independent if the list $v_1, v_2, v_3, v_4$ is....

Problem (a) Show that if we think of $\mathbb{C}$ as a vector space over $\mathbb{R}$, then the list $(1 + i, 1 - i)$ linearly independent. (b) Show that if we think of $\mathbb{C}$ as a vector space over $\mathbb{C}$, then the list $(1 + i, 1 - i)$ linearly dependent. Solution (a) if we let $x,y \in \mathbb{R}$, then $$0 + 0i = 0 = x(1 + i) + y(1 - i) \stackrel{P1A6 \land D1....

Problem Verify the assertion in the second bullet point in Example 2.20 Solution $(\leftarrow)$ This is bullet point number 1 $(\rightarrow)$ If $(2,3,1), (1,-1,2), (7,3,c)$ is linearly dependent, then by definition 2.19 there must exist $x,y \in \mathbb{F}^3$ such that: $$x(2,3,1) + y(1,-1,2) = (7,3,c)$$ From the first and second coordinates above, we see that $2x + y = 7$ and $3x - y = 3$. Adding these we get $5x = 10$ i....

Problem Find a number $t$ such that $$(3,1,4), (2,-3,5), (5,9,t)$$ is not linearly independent in $\mathbb{R}^3$ Solution If we want the given list to be linearly dependent, then we want to find $x,y \in \mathbb{R}$ such that: $$x(3,1,4) + y(2,-3,5) = (5,9,t)$$ From the first two coordinates we see that $3x + 2y = 5$ and $x - 3y = 9$. By adding $-3$ times the second equation to the first, we see that $11y = -22$ i....

Problem Verify the assertions in Example 2.18 Solution $av = 0 \stackrel{P1B2}{\iff} a = 0 \lor v = 0$ , thus $a = 0$ if $v \neq 0$ and if $v = 0$, then $v$ is linear dependent by definition 2.17 Suppose the list $v,u \in V$ is linearly dependent, then there exist non-zero $a_1, a_2 \in \mathbb{F}$ such that $a_1u + a_2v = 0$, thus $u = -\frac{a_2}{a_1}v$....