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Determine the dimensions of Nul $A$ and $\mathrm{Col} A$ for the matrices shown in Exercises $13-18$ .$$A=\left[\begin{array}{rrrrr}{1} & {-6} & {9} & {0} & {-2} \\ {0} & {1} & {2} & {-4} & {5} \\ {0} & {0} & {0} & {5} & {1} \\ {0} & {0} & {0} & {0} & {0}\end{array}\right]$$

The dimension of the column space is 3The dimension of the subspace Null A is 2

Calculus 3

Chapter 4

Vector Spaces

Section 5

The Dimension of a Vector Space

Vectors

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the question asked to determine the dimensions of null A and call A for the Matrix shown below, so as a reminder to find the dimensions of calm. A, um, all you need to do is find the dimension in which thesis abs, space spans or, in other words, find the number of linearly independent columns in the subspace on or, in other words, the number of fact there's in basis. So looking back at the Matrix, we just need to look for the linearly independent columns. And in order to do that, we have to reduce this special inform. But lucky for a surgery is the national in form, so we could just count the pivot columns. So in the first column, we find Pivot Column going down. Then, right there's another pillow pivot going down, and then right there's another pivot, and so far there on Lee three pivot columns. So the dimensions of calm space a equals three. On the other hand, to find the dimensions of null space a being all we have to do is count the number of a free variable columns because, as a reminder, the null space of A is just a set of solutions of the homogeneous equation. Um, a X equals zero. So when we reduce this to reduce ash on farm and we, um, right the Parametric vector form, we get the free variables as part of the its own column. So, for example, writing in Parametric form, the few variables would end up being, Let's say, X three, uh, Times column, plus explore times a column and the number of, um, the number of free variable columns determines the dimensions of no space. A going back to our problem. We find that the third column and the fifth column contain are free variables, and there are only two columns of free variable, therefore, the dimensions of no space a equals two.

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