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Suppose 7*11 matrix A has seven pivot columns: Is Col A = R7? Is Nul A = R4? Explain your answers.Is Col A=RT?0 A No_ Since A has seven pivot columns_ dim Col A =7....

Question

Suppose 7*11 matrix A has seven pivot columns: Is Col A = R7? Is Nul A = R4? Explain your answers.Is Col A=RT?0 A No_ Since A has seven pivot columns_ dim Col A =7. Thus, Col A is seven-dimensional subspace of R" so Col A is not equal to R7 No, Col A is not R7 Since A has seven pivot columns_ dim Col A =4. Thus, Col A is equal to R4_ Yes. Since A has seven pivot columns, dim Col A=7 Thus Col A is a seven-dimensional subspace of R" Col A is equal to R7. No, the column space of A is not

Suppose 7*11 matrix A has seven pivot columns: Is Col A = R7? Is Nul A = R4? Explain your answers. Is Col A=RT? 0 A No_ Since A has seven pivot columns_ dim Col A =7. Thus, Col A is seven-dimensional subspace of R" so Col A is not equal to R7 No, Col A is not R7 Since A has seven pivot columns_ dim Col A =4. Thus, Col A is equal to R4_ Yes. Since A has seven pivot columns, dim Col A=7 Thus Col A is a seven-dimensional subspace of R" Col A is equal to R7. No, the column space of A is not R7_ Since = A has seven pivot columns, dim Col A =0. Thus Col A is equal to 0_ Is Nul A= R4? OA No, Nul A is equal to R4_ Since A has seven pivot columns, dim Nul A = 0. Thus Nul A is equal to 0. No, Nul A is not equal to R4 It is true that dim Nul A = 4, but Nul A is a subspace of R11_ No, Nul A is not equal to R4 Since A has seven pivot columns dim Nul A=7- Thus, Nul A is equal to R7 _ Yes, Nul A is equal to R4. Since A has seven pivot columns, dim Nul A=4. Thus, Nul A is equal- to R4



Answers

(a) Prove that if $a d-b c \neq 0,$ then the reduced row echelon form of
$\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \quad$ is $\quad\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ $r$
(b) Use the result in part (a) to prove that if $a d-b c \neq 0,$ then the linear system $$ \begin{array}{l} a x+b y=k \\ c x+d y=l \end{array} $$ has exactly one solution.
(d) A homogeneous linear system in $n$ unknowns whose corresponding augmented matrix has a reduced row echelon form with $r$ leading I's has $n-r$ free variables.
(e) All leading 1 's in a matrix in row echelon form must occur in different columns.
(f) If every column of a matrix in row echelon form has a leading 1, then all entries that are not leading I's are zero.
(g) If a homogeneous linear system of $n$ equations in $n$ unknowns has a corresponding augmented matrix with a reduced row echelon form containing $n$ leading I's, then the linear system has only the trivial solution.
(h) If the reduced row echelon form of the augmented matrix for a linear system has a row of zeros, then the system must have infinitely many solutions.
(i) If a linear system has more unknowns than equations, then it must have infinitely many solutions.

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.

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's, which are generally independence E c the e f g h by Jay tape. So it'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's with the dimensions off the most waste for a it is beautiful. So therefore, dimension with no space Okay, Is this simply one

Way need to apply. Just urine. Four Queen 9.1. So it is the ring, um, to rink nal ity serum. So it stays for any A is m by and the rank of A Because the minority of a it's the culture in into this province It gave up its condition. That's the common space. So that there Colin Space, it's you co two than now space So they aren't you go. So if there you go one thing is the dimension of the Colin space this Nico to the dimension of the now space. Right? So the dimension of the constipated should go to the rank of A into by definition, the dimension of the now space. It's really equal to the reality that a so here this end of this are the same number. Say, if this is X and this will be X, then end will be two x considered the Dimension X. Here is there is the interchange. So again, it's two times I muted. So Ben is even en If even the any it even means there Colin Number is even so a have even college

In this problem of determinants solution of linear system, we have given that the determinant of a three by three matrix, this is a three by three matrix they didn't mind over three with three medals can also be found using the method of diagonals. So in diagonals we have to write in the form of like this. So this is the metrics. See a 11 A 2, 1 and 831. That's what we a This will be here 1, 2, a 22 and a 32 and this will be a 13 that's what we A 2, 3, and that's what we ate 3, 3. So what we have to do is Mr1, we have to rewrite column one and 2 automatic. Say to the right of metrics, so that's what we hear A 11 column, a 21 and a 31 And now this will be second column that they want to 8 2, 2 and a 32 enough. What we have to do is identify the diagonals. Divan Through the six. So that's what we like this. This will be the first to diagonal. This will be second diagonal. So to say this is the one this would be needed to. And now this would be literally similarly here in this portion, this will be the fourth diagonal. So that's what we're here for. Two diagonals. This will be 50 diagonal. And now this would be the 6th diagonals. And now we have identified it. So here are six and a little and now we have to find these summer products D. One, D. Two, D. Three. So D. One D. Two D. Three. So this will be Divan plus D. Two plus D. Three. That means even +18228337 to 8 to 2 to +3831 like this. And then the step four subtract the some of their products, The 45 basics from the some. So we have to subtract the submission of the food. D. five and d. six From this. d. one d. 2 d. three. So from this we can see that he had each day is a project. So this will be like this 811 8 to 2 833 plus a 1 to 8 to three and +831 plus a 13 A 21 and 832 So this is the one day 23 minus. This will be the value of determinant. A. This will give minus before. So therefore would be a 3182273. So this would be 8318 or two and even three Bliss A 3-8-3. Even one 83 to 8 to three and a 11 plus. This will be a 338 to 1 and a 12 So that's what we ate quickly. E to one and a 12 So when we see it this is the same formula that we have discussed earlier to find the value of determinant. A. So this is the same. So we can say this is the Same as the previous one. So we can say that we can also find the value of determinant by using the diagnosis method.


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