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Finding Coordinate Matrix In Exercises 5-10, given the coordinate matrix of x relative to a (nonstandard) basis B for R", find the coordinate matrix of x relat...

Question

Finding Coordinate Matrix In Exercises 5-10, given the coordinate matrix of x relative to a (nonstandard) basis B for R", find the coordinate matrix of x relative to the standard basisVE8. B = {(.33.(3.4,3). (-}, 6,2}}.[xJBK(0.,0.0 eMr0; [Finding Coordinate Matrix In Exercises 11-16, find the coordinate matrix of x in R" relative to the basis B12. B' = {(-5,6), (3, -2)}, x = (-17,22)

Finding Coordinate Matrix In Exercises 5-10, given the coordinate matrix of x relative to a (nonstandard) basis B for R", find the coordinate matrix of x relative to the standard basis VE 8. B = {(.33.(3.4,3). (-}, 6,2}}. [xJB K(0.,0.0 eMr0; [ Finding Coordinate Matrix In Exercises 11-16, find the coordinate matrix of x in R" relative to the basis B 12. B' = {(-5,6), (3, -2)}, x = (-17,22)



Answers

In Exercises $5-8,$ find the coordinate vector $[\mathbf{x}]_{\mathcal{B}}$ of $\mathbf{x}$ relative to the given basis $\mathcal{B}=\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\}$
$$
\mathbf{b}_{1}=\left[\begin{array}{r}{1} \\ {-2}\end{array}\right], \mathbf{b}_{2}=\left[\begin{array}{r}{5} \\ {-6}\end{array}\right], \mathbf{x}=\left[\begin{array}{l}{4} \\ {0}\end{array}\right]
$$

In his stroller were given a basis B and B No d lectures that that face is so we have the one we have be too and we have B feet and we're us to find the change of ordinance matrix. So we know that in a vector, X is equal to PB times the coordinate vector and this PV is the change of ordinance. Man picks on PB is actually the augmented mentions on by the factors that forms basis beat So pv will be b one b two b three and it is be negative. 1420 negative 58 negative to seven

Okay, We have given be one equal stove one minus one today and B two equals two minus 349 And the trick was to Tu minus 24 And the solution X is 89 six and we have to find Coordinate, Rector. Okay, so we will solve it by be even We do B three Sequels, Toe X. Okay. So we can revive the execution like one minus terry toe and minus one for minus toe minus 394 And this will be aired minus nine six. Okay, so we have to solve this for solving days. We'll go first in r two equals tow R one plus R two. So what we get is one minus tree, too air. This will be zero. This will be one. This will be zero on This will be minus one. Okay. And this will be saying minus 3946 And now we will do what we will do. R three equals toe three r, one plus artery. Then we get one minus story too. And 010 minus one 00 10. 30. Okay. For which we can easily say that. Then she three equals to 30. It means si three equals 23 and C two equals two minus. And for solving this for their we can say see, even minus three. C two plus two equals to eight. And so even plus three plus six equals toe eight. So see, even will be eight minus nine. That is minus one So we can get. Finally, next week we'll store X base B was to C one, C two and C three and that will be minus one minus 13 And that is over. Final answer. Thank you.

In this example, we're starting off with a set B, and we're told that this set forms a basis for our to let's label these two basis vectors as B one and B to for this sake of convenience. Next, we're going to introduce a vector X and said X equal to, let's say, negative two and one. Then, given this vector X, our goal is to find the cornet vector of X relative to the basis be that's given. So if we could determine what this is equal to, our work is going to be complete. But let's start by writing by definition, that this corn in vector would be equal to see one. Let's use a different color here. See one times c two. My eyes will close at the blue bracket. So this is the situation that we start with, and by the definition of a coordinate vector, see one times be one plus C two a times B two must result in the vector X, so that's by definition of having a coordinate vector. And now that we have a vector equation, notice that we can immediately skip to a an augmented matrix where the first column comes from B one, so it's one negative. Three. The second column of this matrix equation comes from B to, so it's too negative. Five. And we're augmenting with the Vector X, which is given to be negative 21 So going to this augmented matrix is going to allow us to solve for the scale. Er's C one and C two weakened by row reduction and I'll copy Row one, which is one too negative, too. Multiply this role one by three at the results of Row two, and we obtain one or excuse me 01 negative five. Then if you copy Row two down, which is euro one. Negative five. Then visualize multiplying row to buy negative two and added to row one we'll obtain one zero and eight. But this then tells us the following vital information the echelon form. Recalling that we've augmented with X and that this court column corresponds to see one and this to see to tells us C one is eight and C two is negative five. So we're ready to stay our solution. The cornet vector of X relative to the basis B is C one, C two, but it solved to be eight n negative five. So what we're saying is, if we put on a tear and a negative five here we have a linear combination that multiplies through resulting in this vector X.

In this problem, we're given a set of basis factor B one b two b p and were given a vector X and where to find the coordinates. Inspector. We know that Vector X will be filled to the purse element of the order Inspector times before, So once you're freeze plus the second l A window before in better times. B +2218 plus the third elemental porting hectic times Be free one. NATO war, too, and X is given here in ecstasy. Quit the Phoenix Fire four. So here, three unknowns He wants to Jesse three End three equations. So the perspiration from for solace or less to see two plus C three is a three second creationists zero plus C two minus C being physical. Five and 30 Creationist three C one plus a C tube plus two C three is equal to four s, so let's must fly the first equation. But a fact you're free and let's use the person 30%. So we have a B C one but 62 plus three C three and a disease of mine. And during a question ist three C one plus 82 for 33 is evil. So let's subtract the 3rd 1 Problem modified for separation. We got negative to see to, uh, plus C three is a fire and our original 2nd 2nd equation issue to minus 33. Is it five? So if you some nose up sea creature like ants love. So we have a gypsy too busy. Zero. So, from this, you know that CTU is zero now, using this, let's find C three. So we have zero minus C three. So from this refined, see three to be fire. Now using 05 Let's find C one from the first equation. Uh, we have C one lost to time. Zero plus five is equal to three. That promise we find See, want to be naked too. So, Dan, the cord inspector is equal to negative to zero


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