determine-the-dimensions-of-nul-a-and-mathrmcol-a-for-the-matrices-shown-in-exercises-13-18-aleftb-3

Question

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

Jillian Rae Villa · Accepted Answer

the question as to determine the dimensions of null space. A in column space A for the Matrix shown below as a reminder of the dimensions of column space A is basically the dimensions in which that subspace fans or the number of the nearly independent columns in the subspace or the number of vectors in its bases, on the other hand, the dimensions of null space. A is just the number of free variable columns and is a reminder. Um, no Space A is a set of solutions of the homogeneous equation. A X equals zero. So once we reduce this to reduce echelon form and right into its parametric form, the columns, the solutions well, the number of columns in thes solutions set will be equal to the number of free variable columns. So going back to our problem, first thing we could do is reduce this to this echelon form. But lucky for us, it&#x27;s already and it&#x27;s special in form, so we can just start counting the number of pivot columns. So, looking at our first column, there is our first pivot column going down into the right. This is our second pivot column, so therefore first column and and the third column, um, our our pivot columns and the&#x27;s are linearly independent, so the dimensions of column space A is the number of linearly independent columns in the subspace, which is to now for the dimensions of no space. A A. We just need to find We just need toe. Find the number of free variable columns, and in this case, the free variable columns are in the second column and in third column, So the dimensions of null space A is the number of free variable columns, which is just to

Runpeng Li · Answer

problems. We&#x27;re keeping a three by three matrix. So that is one or collective 107 zero. Okay, so, um, we can first observe how many people call him. So we&#x27;re there. Be So there are one and two conway on Colin to so that that is too so about free variables. We first have seven of x two Cyril and x one plus four x two minus x three. It&#x27;s cereal, so that gives X one can be determined by X three. So that is X one equals x three and x two is always do so free. Better balls. We have only one, so no a it&#x27;s one.

Runpeng Li · Answer

So in this problem, we first need to find find Ah, number three variables forgiven me tricks. And then we need to find the number of people columns up there. Keep meeting. So, um, seems to give them a trick. Please is already, uh, it&#x27;s already ain&#x27;t the real reduced form. So we can Well, I will persist right down the matrix. Just just for information. This will be, um 13 inactive for two, 21 81 sticks and 00 one 93 70 And there was Hero one. Sir, we have three zeros for this. What? They call him up with 1/3 row. That is 000 14 Letter three and the last rule his whole zeros. Okay, so for this giving matrix, we can find that the people call him ISS. The first column They&#x27;re calling and forth color. So that would be Call him a his three. Okay, so what about the number three variables here? So check the three variables we have. First, there are six variables here. We first have X five plus four us for X six plus. Sorry, Linus. Three x so well, Yeah, We start with the export. Sorry should be explore plus or x five minus three x six, which is cereal. Okay, so from the second rule, we have x three minus three. Explore us seven. It&#x27;s five is zero and from the first rule, we have x one plus three extra checks, too. Minus four x three us two x four minus minus one. Thanks, fi plus six x six. Okay, we&#x27;ll do this step a step. So from this first question we have explore will be determined by x five and experience. So So we know, we know first X five and x x six are free variables. So what&#x27;s more, we have x three be determined by explore next fight, but export can be determined by X files. Maggot never have six. So at three can be determined by the pre variable X five with six. So So we&#x27;re done for the second question and for the first equation, we have x three X for X 56 are all it will be done and next to cannot be done by x five and excess exit. So we have to set It&#x27;s too as a three variables. So this is our set off three variables. So what? That means no off, eh? ISS three

Runpeng Li · Answer

a queue of matrix is two by two matrix, which is three four connective, 6 10 Okay, so, in order to determine the dimensions, um, first transformed this matrix thio reduced ashen of form. So this will be three. I&#x27;m sorry. Three or and we use second role. Plus, why sell the first room? So that would be zero. I saw a bit of personal eight. 18. So that gives X two. It&#x27;s cereal and X one could be ex want us or ex too, is you? So that means everything is cereal. Because we have X to zero and plugging into this equation, we have x one will be zero to so no A it is zero. What about a column? Obey. This dimension will turn out to be, uh, too, because we have two columns, have two columns with the people positions. So we have two people columns so that that is too

Thomas Emment · Answer

case. We want to determine the dimensions off this matrix on dimensions. Matrix are It&#x27;s rose on dhe, its columns. So first off was done with the rose. How many we got? 123 So three rows on. Now he&#x27;s finally columns we&#x27;ve got Well, I&#x27;ve got 12344 columns. So this is gonna be a three by four matrix.

Doruk Isik · Answer

in this problem. We&#x27;re given the system, and we&#x27;re has defied the basis Where d No space. So let&#x27;s say that the silver metric is our eight minutes and then release reporter basis for now. And so we&#x27;re actually looking First system of the question for X is equal to zero. So let&#x27;s write our metrics and destroy it. And that is 100 010 negative, five negative for zero 00176 names. Three. So let&#x27;s write this system in rodeos, reduce or like more as a system up in your question. So in the first row gives us that extorted 65 X tree minus seven. It&#x27;s five second drug users in the question off form extreme physical before ex excusing before three months. Six expire, and the less raw is the same person. Thanks for is equal to be ex, as you can see X three, and it&#x27;s five are cream breakables. So let&#x27;s write our expect. Er, the next will be could do explode next to X tree. Thanks for an expired, and that is, he could through x three bye for 100 plus expire negative seven negative six 03 and one. So these vectors by for one years Europe and negative. 692 Thank you, son. Objectives is 031 are the vectors, um that our basis for no Hey.

Problem

Determine the dimensions of Nul $A$ and $\mathrm{…

Problem 15 Easy Difficulty

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

Answer

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David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

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Linear Algebra and Its Applications