Determine whether given statements about independent vectors are true or false

by RoRi

May 3, 2016

Consider two linearly independent vectors $A,B \in \mathbb{R}^3$ . Determine whether each of the following statements is true or false.

The vectors
$A + B, \qquad A - B, \qquad A \times B$

are linearly independent.
The vectors
$A + B, \qquad A + (A \times B), \qquad B + (A \times B)$

are linearly independent.
The vectors
$A, \qquad B, \qquad (A+B) \times (A-B)$

are linearly independent.

Proof. Consider the equation
$a (A+B) + b(A-B) + c(A\times B) = O \quad \implies \quad (a+b)A + (a-b)B + c(A \times B) = O.$

But, by Theorem 3.13 (page 484) we know that if $A$ and $B$ are independent then $A,B, A \times B$ are independent. Hence, this second equation is true only if $a+b = a-b = c =0$ . But, $a+b = a-b = 0$ implies that $a = b = 0$ . Hence, we have $a = b = c = 0$ , establishing the linear independence of $A+B, A - B, A \times B. \qquad \blacksquare$
Proof. Consider the equation
$\begin{align*} && a(A+B) + b(A+(A \times B)) + c(B+ (A\times B)) &= O\\ \implies && (a+b)A + (a+c)B + (b+c)(A \times B) &= O. \end{align*}$

Since $A,B,A \times B$ are linearly independent (by part (a)) we have

$a + b = a + c = b + c = 0.$

But these equations are only satisfied for $a = b = c = 0$ . Hence, this establishes the linear independence of $A+B, A + (A \times B), B + (A \times B). \qquad \blacksquare$
Proof. Consider the equation
$\begin{align*} && aA + bB + c ((A+B) \times (A-B)) &= O\\ \implies && aA + bB + c ((A \times A) - A \times B + B \times A - B \times B) &= O \\ \implies && aA + bB + 2c(B \times A) &= O. \end{align*}$

But from Theorem 13.13 (page 484) we know that if $A,B$ are independent, then $A,B, A \times B$ are independent. Thus, we must have $a = b = c = 0$ , establishing the independence of $A,B,(A + B) \times (A - B). \qquad \blacksquare$

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