4

Problem 9.1.6. Show that the following row vectors are linearly dependent: (1,1,0), (1,0,1), and (3,2,1). Show the opposite for (1,1,0), (1,0,1), and (0,1,1)....

Question

Problem 9.1.6. Show that the following row vectors are linearly dependent: (1,1,0), (1,0,1), and (3,2,1). Show the opposite for (1,1,0), (1,0,1), and (0,1,1).

Problem 9.1.6. Show that the following row vectors are linearly dependent: (1,1,0), (1,0,1), and (3,2,1). Show the opposite for (1,1,0), (1,0,1), and (0,1,1).



Answers

(a) Show that the vectors $v_{1}=(1,2,3,4), v_{2}=(0,1,0,-1)$ and $\mathbf{v}_{3}=(1,3,3,3)$ form a linearly dependent set in $R^{4}$ (b) Express each vector in part (a) as a linear combination of the other two.

Determine a linearly independent set of vectors that spends the same subspace of V as that spend by the original set of vectors. Well, I'm thinking there's a really good chance that that original set of vectors spans all of riel three dimensional space. So if we find any three of those vectors that also spent all of real three dimensional space, then we will have our our solution. So I'm going to try the 1st 3 vectors. Get the determinant here, just make sure you write it down. Write 1111 negative 111 negative 31 I wrote them as column in the columns column. Vectors, um, one negative 111 Okay, lets calculate the determinant. Negative one negative. Three one negative, one negative three one one is a negative three plus 30 one plus one minus one minus one is also zero, so that may not be the correct choice. Let's try the 1st 2 and the last one. If this doesn't work, we'll have to try a different method first to and the last 1312 That's what we got here. Negative, too. Oops. I forgot to write 1st 2 columns over again. Okay, so that 1st 1 was, um, negative, too. One three minus minus negative. Three. Which is positive three one to Okay, so 36 to I'm thinking that that gives us two. But let me just make sure I didn't make any mistakes. 1111 negative. 11312 negative. 213 plus three, minus one minus two. So, again, 36 to 3 to 2. And so, um, if we choose this this and this, it will spend all of real space. So that is our answer. Uh, those three vectors, this one will be seen as many air dependent or the as the one that we can remove.

In this problem, we have been asked to show that for any vectors U. BMW. In a vector space, the vectors you mine as we the minus W, n W minus. You form a linearly dependent set. Now, if we can show that one of these vectors can be expressed as a linear combination of the other two vectors, then we would have proved that these three vectors form a linearly dependent set. Now, for that let us consider the linear combination of the first two victims as -1 times the first vector -1 times the second vector. Then what we have is minus you, times minus one times minus V. So that is plus, we -1 times. We use minus wheat and then -1 times -W. So that is plus. So we can cancel out the plus ve and the minus V. And we are left with minus you plus W, or w minus you. And this w minus you is the third vector those w minus. You can be written as a linear combination of the first two vectors. And hence this set of three vectors will be a linearly dependent set.

To find the basis of the subspace of R4 spanned by these four vectors. We just put the four vectors into a matrix where each vectors a column of the matrix. And we find the column space of said Matrix. So to find the columns face, we take our matrix and we do Gaussian elimination. So our first row stays the same. And then we're gonna subtract the first row from the second, giving us a 00 two, negative three. And then I'm going to take our second row or a third road to be the same. And then for our fourth row, I'm going to take the fourth row minus the third row, which is going to give us a zero zero zero negative three. Well, at this point, I see that I can scale this negative three in the last row just to be a one. All right. And then I'm gonna continue my Gaussian elimination. I'm first gonna switch the second and the third row. And then I'm going to use the third road to zero out the third column. So I'm gonna just add the first and the third column together. I'm going to take the second row and subtract the third column. Just keep the 3rd column the same. And at this point, I can see in row echelon form that each of my variables are literally independent, which means all four columns are literally independent. So that means the column space is just the collection of my four original vectors or the set right here.


Similar Solved Questions

5 answers
Step Step Substitute Initial J9 1! Use the 331 equilibrium the ICE - table table_ values { Into equilibrium equationStee th Zhwrte al value balanced calculate cquatlon: the molar solublllty of BaFz with Ksp ~ {Step What 8 uonenba 1 this prob
Step Step Substitute Initial J9 1! Use the 331 equilibrium the ICE - table table_ values { Into equilibrium equation Stee th Zhwrte al value balanced calculate cquatlon: the molar solublllty of BaFz with Ksp ~ { Step What 8 uonenba 1 this prob...
5 answers
Gueten 150470 Step 1 d1315.6.20Clauate the sample vamuanceand umplc standard dcyubcn far the fallowlng frequency distribution : heights" centimetets yeer-old boys_If necessary; round to onc Tore decimal place than the lurgest number of decimal places Grven the duta.gmpleol&Heights In CentlmatersClastFrtquenay[18.4 1214[21,5.124.8124.9 . 128 2128. 3 131.613,7 . 135.0Copy Dutu
Gueten 15 0470 Step 1 d1 315.6.20 Clauate the sample vamuanceand umplc standard dcyubcn far the fallowlng frequency distribution : heights" centimetets yeer-old boys_If necessary; round to onc Tore decimal place than the lurgest number of decimal places Grven the duta. gmpleol& Heights In C...
5 answers
Which chemical reaction is consistent with this data presented in the [2] 'graph below? (a) R ~ P(b) R ~ 2 P (c) 2R v P(d) R v3Ple) P - 2R|Time (s)
Which chemical reaction is consistent with this data presented in the [2] 'graph below? (a) R ~ P(b) R ~ 2 P (c) 2R v P(d) R v3Ple) P - 2R | Time (s)...
5 answers
MeasurementJeryMultiplication and division of MeasuremenisDecide whether each proposed multiplication or division of measurements possible_ Ifit is possible_ write the result in the last column of the table_proposedIs thls multipllcatlon or dlvlslon possible?resultOxcD430. JL0.080,60uo(6.0 kg)- (5.0 m)3.0 tL ) (40 mL) =
Measurement Jery Multiplication and division of Measuremenis Decide whether each proposed multiplication or division of measurements possible_ Ifit is possible_ write the result in the last column of the table_ proposed Is thls multipllcatlon or dlvlslon possible? result Oxc D4 30. JL 0.0 8 0,60 uo ...
5 answers
One person speaks with a level of 5Odb.how many people would need to create 65 db. Show your work
One person speaks with a level of 5Odb.how many people would need to create 65 db. Show your work...
5 answers
Consider the indefinite integral gec" (2) tanl (x) dc:Start by setting aside oneThen, apply one of the Pythagorean Identities sec? (2) = 1 + tan? (2) tan- 12(2) sec? (2)Now; let &thenduPerforming the substitution yields the integraldu(Your answer is polynomial in U, written in expanded form)
Consider the indefinite integral gec" (2) tanl (x) dc: Start by setting aside one Then, apply one of the Pythagorean Identities sec? (2) = 1 + tan? (2) tan- 12(2) sec? (2) Now; let & then du Performing the substitution yields the integral du (Your answer is polynomial in U, written in expan...
5 answers
13. The duration of human pregnancy; X, is a normally distributed random variable with mean 266 days and standard deviation 16 days. a. What percentage of pregnancies should last between 270 and 280 days?b. At least how many days should the longest 25% of pregnancies last?Let Y be the number of pregnancies in a sample of 12 that last between 270 and 280 days: Name the distribution of Y and its parameters.d. Calculate P(Y22)
13. The duration of human pregnancy; X, is a normally distributed random variable with mean 266 days and standard deviation 16 days. a. What percentage of pregnancies should last between 270 and 280 days? b. At least how many days should the longest 25% of pregnancies last? Let Y be the number of pr...
5 answers
3)Evaluate the following integral using trig substitutionx3V1-xz dx
3) Evaluate the following integral using trig substitution x3V1-xz dx...
5 answers
-/1 Points]DETAILS SPRECALC7 2.1.059, Find the domain of the function: (Enter your answer using Interval notatlon ) f(t) Vt+ 2
-/1 Points] DETAILS SPRECALC7 2.1.059, Find the domain of the function: (Enter your answer using Interval notatlon ) f(t) Vt+ 2...
5 answers
Position of the boat: Re (c) [5 pts] Find the maximum aid minimum values for the appropriate method. member, YOu are On the interval [0, 10]. Use the
position of the boat: Re (c) [5 pts] Find the maximum aid minimum values for the appropriate method. member, YOu are On the interval [0, 10]. Use the...
5 answers
INTEGRATICN B: FAHTS Given tat ed (untice a1 :, integration ty Farte Icrzula 03=t cha folkwing '~cgrale. Cotpalo ult & ud It' $tet sorreaponding derive jves, th? Chzac: arpmpriately 821 2rsinf-); ')f re*&Pble)r
INTEGRATICN B: FAHTS Given tat ed (untice a1 :, integration ty Farte Icrzula 03=t cha folkwing '~cgrale. Cotpalo ult & ud It' $ tet sorreaponding derive jves, th? Chzac: arpmpriately 821 2 rsinf-) ; ') f re*& Pble)r...
5 answers
Rot Compolnd babre comoareIntcenrantnn end Ostem e 1natat MoN aro nomoicpk dnmnborokg dusioroctonkSelscton norooocenanildDotdeaeteorak
Rot Compolnd babre comoare Intce nrantnn end Ostem e 1natat MoN aro nomoicpk dnmnborokg dusioroctonk Selscton noroooc enanildDot deaeteorak...
5 answers
Pointichargerq +5.0 nclis at the centre ofitwo thinl coricentric conducting spherical shellstol Tradil a cmiand b 6 cm, and charges Qa land Qb respectively: If Qa 42,0 ncand the electric potential at D (rD 5 crm) IS VD 0lidetermine the electric potential VPal PPE#B cm)il(We take V 0 at 0.)PvSelect ane: Olv690.0iv960.09610 V69,0 V
pointichargerq +5.0 nclis at the centre ofitwo thinl coricentric conducting spherical shellstol Tradil a cmiand b 6 cm, and charges Qa land Qb respectively: If Qa 42,0 ncand the electric potential at D (rD 5 crm) IS VD 0lidetermine the electric potential VPal PPE#B cm)il(We take V 0 at 0.) Pv Select...

-- 0.020454--