# Finding transformation matrix of two bases

• Apr 11th 2011, 03:14 PM
Inf
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?

Cheers,
Inf
• Apr 11th 2011, 04:36 PM
Deveno
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 :)
• Apr 12th 2011, 08:39 AM
Inf
Quote:

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?

Quote:

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?

Quote:

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)?
• Apr 12th 2011, 11:26 AM
Deveno
Quote:

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.

Quote:

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".

Quote:

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")
• Apr 12th 2011, 04:42 PM
Inf
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
• Apr 12th 2011, 05:30 PM
Deveno
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.