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If $A$ is a $6 \times 4$ matrix, what is the smallest possible dimension of Nul $A ?$

$N u l A=0$

01:52

Runpeng L.

Calculus 3

Chapter 4

Vector Spaces

Section 6

Rank

Vectors

Campbell University

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

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So in order for us to determine what this ball is possible dimension of Knoll a could be, um we can use the following so we can use the fact that we know the ring. So how do you still drink? Rank of a plus? The dimension of the knoll of a should be equal to S O over here. If this is m by in, it should be equal to end. So if we can figure out well, what should our maximum rank of the speech then we can use that to find our Matt or minimum dimension because Max Rank is saying the same thing as minimizing are no a dimension. Let's go in and just draw six by four matrix. So 123456 Um, so six rows and then four columns everywhere with the red and green light intersect each other we have on entry. So the maximum rank we could have for this is going to be where we have four pivots here. So we have a pivot in each of the columns, so that implies that Max Ring is going to be or eso if we use that appear. So that's gonna be four. Plus the dimension of Noel A is equal to well, in this case in its four. And well, if we subtract that over, that implies our minimum dimension of no a can be zero. So I should probably say men here. Yeah, so that is going to be the smallest possible that we can have.

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