I thought this problem was pretty straightforward, but I can't seem to match the answers in the back of the book.
The problem is: Find the determinant of the following linear transformation.
T(v) = <1, 2, 3> x v (where the x means cross product)
from the plane V given by x + 2y +...
A set of 1000 cards, numbered 1 through 1000 are distributed among 1000 people. Compute the expected number of cards that are given to people whose age matches the number on the card.
So the relevant formula is expected value, but its conditioned because of the ages. So I assumed that the...
To HallsofIvy,
So you're saying we take each basis, perform the linear transformation on the basis, and then we find a linear combination of the original basis that satisfies the transformation and that gives me my column?
So the second column would be something like this
T(1-i) = (1 +...
Well the definition of a transformation is it's closed under addition and scalar multiplication, but I don't see how that helps. Is this regarding finding the matrix of the transformation?
Thanks! But the problem is
Find a basis of Rn such that the matrix B of the given linear transformation T is diagonal.
Reflection T about the line in R^3 spanned by
(1 1 1)
I accidentally put plane! I'm sorry.
So I would find 2 vectors perpendicular to the line and 1 parallel? I...
The problem is
T(x + yi) = x - yi
Show that this is a linear transformation and find the matrix of the transformation using the following basis
(1+i, 1-i)
ARGH
I am having trouble with the complex numbers for some reason!
To show that it is linear I have to show
T(x + yi...
Thanks, I appreciate your help. Now, what is it was a reflection instead of a projection? Let's say a reflection about the plane (1 1 1) Would I do the same procedure?
Find a basis of Rn such that the matrix B of the given linear transformation T is diagonal.
Orthogonal Projection T onto the line in R^3 spanned by
(1 1 1)
I'm assuming (though I tend to be wrong) that I need to find a vector that is parallel to the line and 2 that are perpendicular to...
Wow, now I feel like a moron.
ha ha ha
Grr...
I got confused when I saw that the transformation went from x + yi to x. :/
Sorry, you're right, the book did say T(5i) = 0 exists, therefore it's not an isomorphism.
I think I get it now
The problem:
T(x + yi) = x
C -> C (Complex Numbers)
Show that the above is:
Linear
Isomorphic
This is what i have for showing it's linear:
T(x+yi + a + bi) = x + a + i(y + b) => T(x+a) => T(x) + T(a)
T(k(x+yi)) = k(x) + k(yi) = T(kx) = kT(x)
I assume that i(y+b) = 0. Is...
I thought I had it exact, but my formatting is all wrong. I kept copying and pasting and realized that I ws calling v's x's. I fixed it. :/ Sorry for the confusion.
Well, I've come up with \overline{x_{1}} + 2\overline{v_{2}} + 3\overline{v_{3}} = 0
And since the first column vector has a 1 coefficient (or can be made that way in any other case), then I solve for
\overline{v_{1}} = -2\overline{v_{2}} - 3\overline{v_{3}}
Then where do I go?
"Do...
Since this is a column of only 1 vector representing a kernel, would this represent a plane? I have the equation as
\overline{v_{1}} + 2\overline{v_{2}} + 3\overline{v_{3}} + 4\overline{v_{4}} = 0
First, I solve for \overline{v_{4}}
So is it
\overline{v_{4}} = c1(4\overline{v_{4}}) +...
The v_{4} seems like
1
2
3
4
since the problem is asking for v_{4} to be written as a linear combination of \overline{v_{1}},\overline{v_{2}},\overline{v_{3}}