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DevinBlade
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]
c. Find the change of coordinates matrix P from B to C. [x]C = P [x]B
There's something I don't quite understand. A basis for a 4 dimensional space should contain 4 elements.
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
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.
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.
In short please make sure you copied the question correctly, as is it doesn't make sense.
this is the question my professor stated word for word unfortunately and i have a hard enough time understanding him as it is lol
In any case to check if a vector is in the span of that set, you can solve a system of equations.
You would use row reduction to verify if it is in the span.
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].
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.
just curious sorry if this question is lame but how did you acquire the results for x1, x2, x3 come from?
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
To rephrase this you are seeing if x is in the image of A consisting of columns of B
Do this using row reduction. I will demonstrate.
alright, by the way abtrehearn thanks for the answer to part a it looks good
Start with the augmented matrix: \[\left[ \begin {array}{cccc} 1&0&0&1\\ 0&1&0&1 \\ 2&3&4&1\\0&0&1&-1\end {array} \right] \]
Now row reduce the matrix.
This would be the row reduced form? 1 0 0 1 0 1 0 1 0 0 1 -1 0 0 0 0
@ sir alchemista please solve my liinear algebra questions
Yes, now do you see that the results of the row reduction yield the coefficients for the linear combination that yield x?
ahh yah just pops right up there aha alrighty thanks to you both ill retype part be and c as a new question XD
But its not magic, you should understand the theory behind it.
yah just didnt realize it before until i was actually looking at the complete form, just something i over looked i guess aha
Right on, DevinBlade :^)
@sir abtrehearn please see my linear algebra questions i posted
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
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.
<<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] >> 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\[z_1b_1+z_2b_2+z_3b_3=c_3.\]
Yep, there is already a solution in the new thread.
http://openstudy.com/groups/mathematics/updates/4e1d5ab10b8b4841c1a9b0d9