- anonymous

Greetings, This question deals with linear algebra so it sucks here it is!
3a. Given the vector x and the basis B = [b1, b2, b3] of a subpace of R4 determine if x is in the span of B, and if so find the coordinates of x with respect to B and give the coordinate vector [x]B.
x:[1,1,1,-1] b1: [1,0,2,0] b2:[0,1,3,0] b3:[0,0,4,1]
b. Given the basis C = [c1, c2, c3], determine if C is in the span of B. If so write the vector x using the C coordinates. and give the coordinate vector [x]C.
c1:[1,2,-4,-3] c2:[1,3,3,-2] c3:[-1,3,-9,4]

- katieb

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- anonymous

c. Find the change of coordinates matrix P from B to C.
[x]C = P [x]B

- anonymous

There's something I don't quite understand. A basis for a 4 dimensional space should contain 4 elements.

- anonymous

the 4th element i believe is x:[1,1,1,-1] the instructor i have now is completely horrible and doesn't even teach any thing.. so i could be wrong lol

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## More answers

- anonymous

The question doesn't make sense unless you have a basis for the vector space. A basis must have the same cardinality as the dimension of the vector space.

- anonymous

Oh and furthermore to ask of a vector is in the span of a basis is nonsensical. Of course any vector in the vector space can be expressed as a linear combination of a basis.

- anonymous

In short please make sure you copied the question correctly, as is it doesn't make sense.

- anonymous

this is the question my professor stated word for word unfortunately and i have a hard enough time understanding him as it is lol

- anonymous

In any case to check if a vector is in the span of that set, you can solve a system of equations.

- anonymous

You would use row reduction to verify if it is in the span.

- anonymous

Part a:
Vector x is in the span of B if and only if it is a linear combination of the vectors in B. Then there are scalars
\[x_{1} , x_{2} , x_{3}\]
such that
\[x_{1} b _{1} + x_{2} b_{2} + x_{3} b_{3} = x,\]or
\[x_1 (1,0,2,0) + x_2 (0,1,3,0) + x_3 (0,0,4,1) = (1,1,1,-1)\]
a system of four simultaneous linear equations in unknowns
\[x_1, x_2, x_3,\]which has the unique solution
\[(x_1,x_2,x_3) = (1,1,-1),\]
so the vector x can be expressed as a linear combination if the vectors in B, and x is in the span of b. The coordinate vector [x]B is [1,1,-1].

- anonymous

To setup this system of equations, start with the vectors in the set B as the columns of a matrix. You "augment" the matrix with the vector x.

- anonymous

just curious sorry if this question is lame but how did you acquire the results for x1, x2, x3 come from?

- anonymous

By solving the system of equations. You are essentially seeing if there exists a linear combination of the vectors in B such that the result of the linear combination is x

- anonymous

To rephrase this you are seeing if x is in the image of A consisting of columns of B

- anonymous

Do this using row reduction. I will demonstrate.

- anonymous

alright, by the way abtrehearn thanks for the answer to part a it looks good

- anonymous

Start with the augmented matrix:
\[\left[ \begin {array}{cccc} 1&0&0&1\\ 0&1&0&1
\\ 2&3&4&1\\0&0&1&-1\end {array}
\right] \]

- anonymous

Now row reduce the matrix.

- anonymous

This would be the row reduced form?
1 0 0 1
0 1 0 1
0 0 1 -1
0 0 0 0

- anonymous

@ sir alchemista please solve my liinear algebra questions

- anonymous

Yes, now do you see that the results of the row reduction yield the coefficients for the linear combination that yield x?

- anonymous

ahh yah just pops right up there aha alrighty thanks to you both ill retype part be and c as a new question XD

- anonymous

But its not magic, you should understand the theory behind it.

- anonymous

yah just didnt realize it before until i was actually looking at the complete form, just something i over looked i guess aha

- anonymous

Right on, DevinBlade :^)

- anonymous

@sir abtrehearn please see my linear algebra questions i posted

- anonymous

Let me see...

- anonymous

thanks very much abtrehearn, im reasking the question for part b so answer it on there i think this question is in deserves of multiple medals :P

- anonymous

last one is here http://openstudy.com/users/devinblade/updates/4e1d5ab10b8b4841c1a9b0d9* you should type the answer there so i can mark it answered but answering it here is fine too.

- anonymous

<>
C is in the span of B if and only if each vector in C can be expressed as a linear combination of the vectors in B. So we seek scalars
\[x_1, x_2, x_3\]such that \[x_1 b_1 + x_2 b_2 + x_3 b_3 = c_1,\]scalars\[y_1, y_2, y_3\]such that\[y_1b_1 + y_2 b_2 + y_3 b_3 = c_2,\]and scalars \[z_1, z_2, z_3 \]such that\[z_1b_1+z_2b_2+z_3b_3=c_3.\]

- anonymous

Yep, there is already a solution in the new thread.

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