Thread: 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

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

Originally Posted by Deveno

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?

Originally Posted by Deveno

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?

Originally Posted by Deveno

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)?

Originally Posted by Inf

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")

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

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.