## anonymous 4 years ago 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]

1. anonymous

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

2. anonymous

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

3. 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

4. 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.

5. 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.

6. anonymous

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

7. 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

8. anonymous

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

9. anonymous

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

10. 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].

11. 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.

12. anonymous

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

13. 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

14. anonymous

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

15. anonymous

Do this using row reduction. I will demonstrate.

16. anonymous

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

17. anonymous

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

18. anonymous

Now row reduce the matrix.

19. anonymous

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

20. anonymous

@ sir alchemista please solve my liinear algebra questions

21. anonymous

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

22. 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

23. anonymous

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

24. anonymous

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

25. anonymous

26. anonymous

@sir abtrehearn please see my linear algebra questions i posted

27. anonymous

Let me see...

28. 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

29. anonymous

30. anonymous

<<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$z_1b_1+z_2b_2+z_3b_3=c_3.$

31. anonymous