Please tell me if you can whiten your teeth by scrubbing them with toothpaste whitener?

Not really: Brushing does not offer enough contact time of a whitening agent for it to effectively whiten teeth and most toothpastes do not contain much more than abrasives. Good oral hygiene will reduce surface stain which may help the teeth look better but does not alter or lighten the actual color, like professional whitening or whitestrips can do.