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For vectors $x, \: y$ in $\mathbb{R}^n$, define the inner product $\langle x, y \rangle = \Sigma^n_{i=1} x_iy_i$, and the length of $x$ to be $\| x \| = \sqrt{\langle x, x \rangle}$. Let $a, \: b$ be two vectors in $\mathbb{R} ^n$ so that $\| b \| =1$. Consider the following statements:

  1. $\langle a, b \rangle \leq \| b \|$
  2. $\langle a, b \rangle \leq \| a \|$
  3. $\langle a, b \rangle = \| a \| \| b \|$
  4. $\langle a, b \rangle \geq \| b \|$
  5. $\langle a, b \rangle \geq \| a \|$


Which of the above statements must be TRUE of $a, \: b$? Choose from the following options.

  1. ii only
  2. i and ii
  3. iii only
  4. iv only
  5. iv and v
in Linear Algebra by Veteran (98.4k points)
edited by | 218 views

1 Answer

+3 votes
Let $n =1$ then,

if $a$ $=$ $\left \{ 2 \right \}$, $b$ $=$ $\left \{ 1 \right \}$

< $a,b$ > $=$ $2$

$||$ $b$ $||$ $=$ $1$

Hence, i is incorrect.

Let a = {-1} , b ={1}

$< a,b > = -1$

$||$ $a$ $||$ $||$ $b$ $||$ $=$ $1$ x $1$ = $1$

Hence, iii and iv is incorrect, if iv is incorrect $d$ and $e$ both can't be right.

 So, by elimination $(a)$ is correct.
by Active (1.7k points)
edited by

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