Linear Algebra 2 - Representing Matrix

In summary, the matrix representing the function L is L[x], where x is the variable being acted upon. The action of the function L is to replace x with x+1, yielding the matrix L[x+1].
  • #1
27
0
Sry, this will be the last question^^

10muavl.jpg


Its a similar to the problem i ve postet before. Maybe if i can solve the first problem i can solve this to. But what i don't understand is that notation. I ve circled it with a red line.
Does anyone know what this means?

Thx
Mumba
 
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  • #2
It means the representation of L in the basis B = {1, x, x²}. They're just calling that matrix AB
 
  • #3
So this A is the representing matrix?
But what does then Ap=q mean? Or how can i calculate L for 1, x and x^2?
 
  • #4
Ap = p(x+1) defines the action of L.

So for p = 1, you get L(1) = x + 1 and so on.
 
  • #5
Actually, it looks like q(x+1) is not intended as multiplication of q with (x+1). The action of L on the function p(x) is given by p(x+1).

So, if p = 1 + x, then L(p) = 1 + (x + 1) = 2 + x.
 
  • #6
But so i get for
L(x) = x^2 + x,
L(x^2)=x^3 + x^2

So should i choose a new basis, for example {1+x,x^2,x^3} to get the repr matrix?
 
  • #7
See my post above.

L(x) will be x + 1 instead of x² + x.
 
  • #8
L(p)=q(p)=p(x+1)?

wie kommst du da auf L(x)=1+x?
Was hast du denn dann für L(1)?
 
  • #9
Treat a polynomial as the function p(x). The action of L is to take that function and return the function q(x) = p(x+1)

So, if p(x) = 1 + x, then q(x) = p(x + 1) = 1 + x + 1 = x + 2 (i.e. L[1 + x] = 2 + x)

If p(x) = 1, then q(x) = p(x + 1) = 1. (i.e L[1] = 1 )
 
  • #10
So i get then
L(1) = 1
L(x) = x+1
L(x^2)=x^2+1

?
 
  • #11
L[x²] = (x+1)² = x² + 2x + 1
 
  • #12
and then for
L(1) --> 1 0 0
L(x) --> 1 1 0
L(x^2) --> 1 0 1

So my matrix would be
1 1 1
0 1 0
0 0 1

??
 
  • #13
ahh ok so if u change this fpr L(x^2) --> 1 2 1
matrix:
1 1 1
0 1 2
0 0 1

correct? ^^
 
  • #14
Yes, that's correct.
 
  • #15
:D:D
cool thanks alooooot
 
  • #16
dx said:
Treat a polynomial as the function p(x). The action of L is to take that function and return the function q(x) = p(x+1)

So, if p(x) = 1 + x, then q(x) = p(x + 1) = 1 + x + 1 = x + 2 (i.e. L[1 + x] = 2 + x)

If p(x) = 1, then q(x) = p(x + 1) = 1. (i.e L[1] = 1 )

Should it be like that:

If p(x) =1 then, q(x) = p(x+1) = p(x)+p(1) = 1 + 1 = 2, so L[1]= 2!?
 
  • #17
No, p(x) is a constant function, i.e. it has the same value for any x. So p(x + 1) would still be 1.
 
  • #18
hmm, ok
but why ist L(x) = x+1?

--> p(x) = x
if its a constant function and p(x)=x, shouldn't be P(x+1)=x too?
 
  • #19
p(x) = 1 is a constant function, p(x) = x is not!

For a given polynomial p, we get the L[p] by replacing every x with x + 1.

So if p = 1, we do nothing since there is no x, and L[1] = 1

If p = x, we replace x by x + 1, and we get L[x] = x + 1

If p = x2, we replace x by x + 1 to get (x + 1)2, i.e. L[x2] = (x + 1)2 = 1 + 2x + x2
 
  • #20
oh man
yes sure thanks again ^^
 

1. What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is commonly used in mathematics, physics, and engineering to represent and manipulate data and equations.

2. How are matrices represented?

Matrices are typically represented using brackets or parentheses to denote the beginning and end of rows and columns. For example, a 3x3 matrix would be represented as [ [a, b, c], [d, e, f], [g, h, i] ] or ( (a, b, c), (d, e, f), (g, h, i) ).

3. What is the difference between a row vector and a column vector?

A row vector is a matrix with only one row, while a column vector is a matrix with only one column. A row vector is written as [a, b, c], while a column vector is written as [a; b; c].

4. How do you add or subtract matrices?

To add or subtract matrices, the matrices must have the same dimensions. This means that they must have the same number of rows and columns. To add or subtract matrices, simply add or subtract corresponding elements in the matrices. For example, to add [ [1, 2], [3, 4] ] and [ [5, 6], [7, 8] ], you would add the elements in the first row of the first matrix with the elements in the first row of the second matrix, and then do the same for the second row.

5. What is the identity matrix?

The identity matrix is a special type of square matrix where all elements on the main diagonal (from top left to bottom right) are equal to 1, and all other elements are equal to 0. The identity matrix is often denoted as I and is used in matrix multiplication as the equivalent of multiplying by 1.

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