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Matrix Image and Kernel 
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#1
May2910, 11:26 AM

P: 51

1. The problem statement, all variables and given/known data
i) Find the Image and Kernel of A = (2,1)(4,2) (where each bracket is a row). ii) Calculate A^{2} and use i) to explain your result. 2. Relevant equations None 3. The attempt at a solution So I can do everything up to the very last bit (i think anyway). i) The Kernel = (1,2) = Image. ii) A^{2} = 0 but this is where I don't know what to say. How do I use part i) to explain the 0 matrix found? 


#2
May2910, 12:39 PM

Mentor
P: 21,214

Geometrically, A maps any vector along the line 2x + y = 0 to the zero vector. A maps any vector x not along the the line 2x + y = 0 to a vector along this line. IOW, if x is not in the kernel of A, A projects it onto this line. For ii, since A^{2}x = A(Ax)think about what A does to a vector x, and then think about what A does to a vector Ax. 


#3
May3110, 03:05 PM

P: 51

OK i think i understand what you mean about the kernel. so applying to to another question, if I have the matrix:
A=[{1,0,2},{2,2,0},{0,3,6}] and I wanted to find the kernel, I'd reduce it down to: A=[{1,0,2},{0,1,2},{0,0,0}] and thus the kernel is: Ker[A] = f[1,1,2] where f is any number. Is that about right? 


#4
May3110, 03:58 PM

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PF Gold
P: 11,692

Matrix Image and Kernel
I think you got the 1's and 2's switched, i.e.
[tex]\textrm{Ker}[A] = \{\vec{x}\in \Re^3\,\, \vec{x} = f(2,2,1), f \in \Re\}[/tex] 


#5
Jun110, 07:09 AM

Math
Emeritus
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PF Gold
P: 39,363




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