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Math Help - Finding transformation matrix of two bases

  1. #1
    Inf
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    Finding transformation matrix of two bases

    Hey guys,

    i tried to figure out a solution for the following
    excercise:

    May B=(b1,b2) an ordered basis of the real vector space V and C=(c1,c2,c3) may be an ordered basis of the real vector space W. The map f:V→W is defined as
    f(b1) = c1 + c2 + 2c3 and f(b2) = 2c1 + c2 - c3.

    Find the matrix that transforms B to C of f.

    Idea:
    X * B = C

    Since B is a 2x1 and C is a 3x1 matrix
    X is supposed to be a 3x2 matrix to transform B to C. Is that right?
    If yes, how to solve this equation system in which i have 6 unknows?

    Thanks in advance!

    Cheers,
    Inf
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  2. #2
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    suppose v in V is any old vector. isn't it true that we can write v = v1b1+v2b2?

    does this become an easier problem if we define v = (v1,v2), relative to B?

    how about if we write f(v) = (w1,w2,w3), relative to C? i don't see ANY variables now, just numbers
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  3. #3
    Inf
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    Quote Originally Posted by Deveno View Post
    suppose v in V is any old vector. isn't it true that we can write v = v1b1+v2b2?
    True, so v1 and v2 are vectors and b1 and b2 are real numbers?

    Quote Originally Posted by Deveno View Post
    does this become an easier problem if we define v = (v1,v2), relative to B?
    This v = v1b1+v2b2 is v defined relativ to B, right?

    Quote Originally Posted by Deveno View Post
    how about if we write f(v) = (w1,w2,w3), relative to C? i don't see ANY variables now, just numbers
    Sry, but now i get confused with variables...
    f ist only defined for b1 and b2 what to write for f(v)?
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  4. #4
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    Quote Originally Posted by Inf View Post
    True, so v1 and v2 are vectors and b1 and b2 are real numbers?
    no {b1,b2} form a basis. the elements of a basis are vectors. v1 and v2 are the coordinates of v relative to the basis B.

    This v = v1b1+v2b2 is v defined relativ to B, right?
    yes. so v1b1+v2b2 = (v1,v2)B. normally, when a vector is written (x1,x2), the basis is understood to be the "standard basis" {(1,0), (0,1)}, but a vector

    need not be defined relative to the standard basis. think of it this way: a vector is itself, sitting out there in space. to write its coordinates down,

    WE pick a basis, the standard basis is the usual choice, it's orthogonal, and easy to work with. in this case, we have a basis, we don't know the standard

    coordinates for them. so we just identify our vectors from their "b1-axis coordinate" and their "b2-axis coordinate".

    (v1,v2)B is v in "B-coordinates".



    Sry, but now i get confused with variables...
    f ist only defined for b1 and b2 what to write for f(v)?
    you only need to know what f is on b1 and b2 to define f, because f is LINEAR.

    f(v) = f(v1b1+v2b2) = v1f(b1) + v2f(b2). so f((v1,v2)B) = ...? (your answer should be in "C-coordinates")
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  5. #5
    Inf
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    Thank you, Deveno!

    Now i understand... i thought b1 and b2 are real numbers and B is a vector... xD

    To hightlight vectors i put them in []-brackets

    v = (b1, b2)B = v1 * [b1] + v2 * [b2]

    f(v) = f(v1 * [b1] + v2 * [b2]) = v1 * f([b1]) + v2 * f([b2])
    = v1 * ([c1] + [c2] + 2[c3]) + v2 * (2[c1] + [c2] - [c3])
    = v1*[c1] + v1*[c2] + 2*v1*[c3] + 2*v2*[c1] + v2*[c2] - v2*[c3]
    = (v1 + 2*v2)*[c1] + (v1 + v2)*[c2] + (2v1 - v2)*[c3]

    c = (v1 + v2, v1 + v2, 2v1 - v2)

    Every vector is a matrix right? so c is the transformation matrix?

    Thanks!
    Inf
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  6. #6
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    in my earlier posts, v is an "example". you have actual NUMBERS for the C-coordinates of the images under f of the basis vectors b1 and b2.

    you should have a matrix made up of numbers.

    f(v) = f((v1,v2)B) = f(v1b1+v2b2) = v1f(b1)+v2f(b2) =v1(c1+c2+2c3) + v2(2c1+c2-c3)

    = (v1+2v2)c1+(v1+v2)c2+(2v1-v2)c3 = (v1+2v2,v1+v2,2v1-v2)C

    but that is only the definition of the linear transformation f. you need a MATRIX.

    what are the B-coordinates of b1? b1 = 1b1+0b2 = (1,0)B <----see? the coordinates are NUMBERS.

    the B-coordinates of b2 are: (0,1)B.

    the C-coordinates of f(b1) are: (1,1,2)C. the C-coordinates of f(b2) are: (2,1,-1). the matrix of f, relative to B and C is a matrix for which:

    [a b][1]....[1]
    [c d][0] = [1]
    [e f].........[2] <----what does that tell you about what (a,c,e) is? there are no "unknowns" or equations to solve, you should be able to

    simply write down the matrix for f.
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