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In Exercises 11 and $12,$ give integers $p$ and $q$ such that Nul $A$ is a subspace of $\mathbb{R}^{p}$ and $\mathrm{Col} A$ is a subspace of $\mathbb{R}^{q}$ .$$A=\left[\begin{array}{rrrr}{3} & {2} & {1} & {-5} \\ {-9} & {-4} & {1} & {7} \\ {9} & {2} & {-5} & {1}\end{array}\right]$$

$p=4, q=3$

Algebra

Chapter 2

Matrix Algebra

Section 8

Subspaces of Rn

Introduction to Matrices

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Okay, Um, let's try this again. It's still doing the same weird thing. Okay, so I have matrix a here. This is a problem. Number 11 from section 2.8. I'm given a matrix A, um and I'm after the null space and the cone space. Okay, so on drily, this is just the null space is just a question of matrix multiplication. So we're after all the vectors accession X equals zero and just examining matrix multiplication. You can see that the vectors X would have to be in our four. So that means that the, um uh the, um que ah que would have to equal to four. Um, this is a subspace of our four. I'm sorry. I don't know what's going on here all of a sudden, um, this is contained in, um are sore. Okay. On dso Similarly, when we look at the column space So we're looking at the span of the columns, and just by looking at the columns, you can see that there three, they have three coordinates. So that means that the column space, um, is contained in our three. So the column space is a subspace of our three And that means that he would have to be, um uh in P would have to be three p. Just being the thes subspace are the dimension of the subspace in which the Colin spaces in. So just by looking at that, you can see that the Colin space is contained in our three. So, uh, yeah, so the null space eyes all the vectors acts such an ax equals zero. And just by looking at the the number of columns in a, you can see that it's four. So that means that it would have to be a subspace of our four. And the column space is just the subspace spanned by the columns. And when you look at the columns, you can see that they're three dimensional vectors. So that means that it would have to be contained in art. So that's it for this problem here. Thank you very much and have a nice

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