suppose-a-4-times-7-matrix-a-has-four-pivot-columns-is-col-amathbbr4-is-nul-amathbbr3-explain-your-a

Question

Suppose a $4 	imes 7$ matrix $A$ has four pivot columns. Is Col $A=\mathbb{R}^{4} ?$ Is Nul $A=\mathbb{R}^{3} ?$ Explain your answers.

Subham Jyoti Mishra · Accepted Answer

Yeah, well, given the metrics air, which is 4 to 7 on our tricks and your tests been Thank you. And and the question that this mattresses were by what columns? So therefore, the diamond Anson column space is your since l 45 columns. Andi, Uh, the question that has been given that we have to find out if the column space is Issa Space off. Uh huh. Who and if in all spaces, a space off our treatment now the first part since Carl, um A is four payments of space off powerful. That is more. There are 45 What elements in call in air So that the dimensional economy is food on hands. It is a subspace apart pores. Therefore, concerts with our work. The worse part. We can agree, uh, call them air. This system is based off powerful Mhm. But the null space off air has diamonds and on minus a rank A which is equal to seven minus any number of columns and seven minus four, which is a good of three. So the diamonds and of knows basis is three. So therefore okay, I think is a three diamonds, another victor space with seven elements since number of columns of seven. So each director has seven elements in it so that our has toe be subspace off, uh, seven. So it cannot be suspects a party.

James Chok · Answer

Okay, So suppose that a is a four by seven. Matrix has four rows and seven columns, which, which has three titted columns, is the calm space. All three. Well, if a has three pivot columns, then it means that the color space is span. The calm space off, eh is span by three Becker&#x27;s, which are generally independence E c the e f g h by Jay tape. So it&#x27;s important to understand that because he has three kids columns 123 columns are linearly independent of each other. And because they exist in our four and they are literally independent of each other, this is offering So the dimensions off the mall space if we get direct knowledge IQ there which say that the rank Oh, aye, plus, by the dimensions off the null space okay, must be must be equal to end which in this case, is for well, we know that the rank is Ah, calm is the number of pivot columns. So we have 33 what&#x27;s with the dimensions off the most waste for a it is beautiful. So therefore, dimension with no space Okay, Is this simply one

Bobby Barnes · Answer

right, So we&#x27;re giving this five by six matrix with four pivot column. So the first thing they say is determined the knoll of So there&#x27;s actually a Thurman book that says, Um, so if we have a Matrix works and buy in here and is supposed to be equal to the rank of our matrix, so rank a or the dimension of the column space, plus the dimension off Knoll A. And from knowing we have poor pivots, that tells us our rank is four and in a six. So this is 62 to 4, plus the dimension off the normal of A and will be subtract four over That gives us two is equal to the dimension of our normal space. So no a. So that answers the first question. Now, whether or not the call of space is the same as our for what we have to remember that our four is supposed to be vectors with four entries. Bucks. Let&#x27;s actually look at what a five by six matrix looks like. So remember this is rose by columns, so we should have six road or five rows and six columns. Um, so it would be a 12345 And then 123456 So this is kind of like what it looks like. And everywhere there was like, an intersection. We have, like, a entry. Well, notice one of these vectors. If we were to look at it in the column space, it has 12345 entries. So this is going to be evolves because, um, the entries of column A are going to be by the dimensional, um, so false sense, all vectors and column of a have five in the Great Hall of the Better five, um, entries. So that&#x27;s how you kind of rationalize this out.

Michael Jacobsen · Answer

in this video, we have a three by N Matrix. So that&#x27;s three rows and columns, and we&#x27;re going to assume that the columns of a are going to span all of the space are three. Next, let&#x27;s addict something extra to this picture. Let&#x27;s say assume de is a three or N by three matrix, for which eight times D equals I three. So what we want to determine here is what does this matrix de equal in particular? Well, let&#x27;s start with I three. We know that I three is the three by three identity matrix, so it is formed by columns E one E two e three, where e one is 100 E two is 010 anti three is 001 so altogether were then looking for a matrix D, such that when we multiply a times d, we obtain this three by three identity matrix we see displayed here. First, let&#x27;s also note that since the columns of a span or three each time we write a matrix equation eight times X equals B, where be is in are three. This matrix equation is always consistent. This is because knowing that the columns of a spin or three is logically equivalent to the statements that a X equals B is always a consistent system of equations. Let&#x27;s use that fact next to construct our matrix de here, we can say that the systems eight times X equals e one eight times X equals e to eight times X equals e three were e one e two e three are the columns of our identity matrix here. Each one of these systems are consistent, since we could have placed anything in these positions on the right hand sides and they&#x27;re still consistent. Let&#x27;s suppose then that let&#x27;s call this d one de to d three our solutions. So in other words, let&#x27;s say what we mean exactly were saying that since eight times X equals e one e two or three here is always consistent. It follows that a Times D one must be equal to the one. Since this is our first solution. Eight times D, too, is E to and likewise, eight times D three is E three. Now let&#x27;s then define the matrix D By columns D one do to d three, where these came from. The solutions that we had constructed. Then when we take a times D, this is what will produce by the definition of a matrix times another matrix. We take a Times first D one, which came from here. Then the second column of the product is a times D to and lastly, we have eight times d three for the third column of this product. So this is what eight times D looks like. But we also know that eight times d produces e one e two, any three so we could make a substitution here, here and here to get the following. We have the one e two and e three, but this is now the three by three identity matrix. So we&#x27;ve just shown how can to construct the particular matrix de such that eight times d equals I three.

Patrick Vaughan · Answer

Hello. So today let&#x27;s do a more theoretical type problem. So let&#x27;s show that a arbitrary three by seven matrix of a, uh, has banality of four and it must have a column space. Let&#x27;s show that it must have a column space, um, where it&#x27;s in the, ah, third dimension and is the road space of a then also in the third dimension. So let&#x27;s review our rules. Rank ability zero. So our rank milady here, um, says that e rank of some arbitrary nature. A plus, the reality, the dimension of in all spaces day equals the number of columns. So in this three by seven matrix, we have three rounds, seven columns that we know the number of columns of seven. We know that the utility we&#x27;re assigning that to four and the rank we don&#x27;t know. Yeah, but using this rule, this tells us that the bank of a equals those seven columns minus the formality that were assigned, so that gives us a record three. So let&#x27;s build this arbitrary matrix. And let&#x27;s just assume that it&#x27;s already in produced garage one form just because that makes everything a little be easier. So we have one to you. Three rows. We have our leading one. Our next one. When I went after that, then we have seven columns. So 12 three for five, 67 And we&#x27;ll just fill battle up zeros to make this nice and easy. So this is our three by sudden Matrix. So and reduce little up. Throw up, Sean form The, uh we have the rank is three. So that&#x27;s the number of tipping point 12 three. And then that told us that the column space is the, uh in the reduced Raj lawn form is the Matrix Farm. But we&#x27;re just assuming about this matrix and the original Given. So not only is it released Rachael on form, it equals our original matrix. So our colons stays is actually going to be these three columns because that&#x27;s where a pivot points are. So the column space is the one. Oops, let&#x27;s undo that. See, one equals one 00 see, to equals euro 10 and C three equals euro 01 So we have three doctors in the column space, so this makes it such that the column space all day equals three dimensions because you could just arbitrarily say that this is you&#x27;re x dimension, or why dimension and your son dimension. So yeah, it must have a concept of that. So let&#x27;s done. Look at the road. Space doesn&#x27;t have to be three dimensions. Or is it three dimension? So the road space of our original matrix, But we just arbitrarily chose Do you already be in this reduced row? Actual on form 1234567 Okay, you&#x27;re 10 There&#x27;s one. So the roast space is the non zero rose. So we look at it. We have one hears looking on zero a non zero and a non zero. So the roast space in land such that. And let&#x27;s do this in blue or green. No one equals one 0 to 3 for 567 wrote to equals zero one 03456 seven And then our road three. I&#x27;m sure you can see where this is going. No. Is zero euro one, 4567 So, again, we can look at this and say that this is our excess. Why space? And once again we see that Yes, indeed, for this three by seven matrix that we arbitrarily chose. That the row space zooming back in. Now we don&#x27;t need to see that the row space, okay does indeed show up in three dimensions.

Thomas Emment · Answer

Oh, you&#x27;re here. We&#x27;ve got given this matrix a on with till we would find me a squared a cubed 84 85 80 square just to be a times by a So we&#x27;ll do Thio going to let you matrix resulting do the first very terms Affairs column one times one is 10 times one is 01 there in the 1st 3 times a second column to get the first the second column point one times 002 times one is once you have zero. Next up for the fest, call him here. Second rose or a second row first column that I dated around 111111112 21 1 plus one is two on then, for the final one will have one time Sierra wants Well, so this is a squared, not get a kit. We&#x27;ll simply after times by again. Okay, so we have, um for the That&#x27;s right. Uh, this column will have 1 to 01 times one. So this is gang anyone, And I think you start seeing a pattern here Next. 11010 zero. Uh, and then we have 211 Once we have a three hand out and then we have 21011 Again. I hope you can see the trend here. I&#x27;m gonna do the next. This is This is now a keep on, then aid to the full was gonna times by a again one. Yeah, on then. Say deed Road times the column Sylvia ones there. There once actually won again. 100101010 Now, I&#x27;ve got three times one or one of the ones we got full. We have three times zero. So this is a full on then? I hope you can see that a five times again. Gonna get 1051 This is gonna be a five. So whatever a to the N is, it&#x27;s going to be equal to one and zero. And that&#x27;s high tech His matrix

Problem

Suppose a $5 \times 6$ matrix $A$ has four pivot …

Problem 7 Easy Difficulty

Suppose a $4 \times 7$ matrix $A$ has four pivot columns. Is Col $A=\mathbb{R}^{4} ?$ Is Nul $A=\mathbb{R}^{3} ?$ Explain your answers.

Answer

Thus, the dimension of null space of the matrix is 3 .
since, the 3 dimensional vectors may or may not belong to $\mathrm{R}^{3}$ .
Thus, the nullity of the matrix is not $\mathrm{R}^{3}$ .

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Thus, the dimension of null space of the matrix is 3 .since, the 3 dimensional vectors may or may not belong to $\mathrm{R}^{3}$ .Thus, the nullity of the matrix is not $\mathrm{R}^{3}$ .

Linear Algebra and Its Applications

Heather Z.

Caleb E.

Michael J.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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

Anna Marie V.

Samuel H.

Michael J.

Joseph L.

Thus, the dimension of null space of the matrix is 3 .
since, the 3 dimensional vectors may or may not belong to $\mathrm{R}^{3}$ .
Thus, the nullity of the matrix is not $\mathrm{R}^{3}$ .

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications