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Topic: rank and column space of a matrix (Read 2459 times) |
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Nikki
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a) suppose that A is a 3*3 matrix whose nullspace is a line thorugh the origin in R3. Can the row or column space of A also be a line through the origin? ExplaiN
b) If A is a 6*4 matrix such that the system Ax=0 has a non-trivial solution, what is the largest possible value of rank (A)? explaiN
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ThudnBlunder
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Re: rank and column space of a matrix
« Reply #1 on: May 27th, 2004, 9:00am » |
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Quote:
Your punctuation is rather poor. You should write "Explain!"
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| « Last Edit: May 27th, 2004, 9:01am by ThudnBlunder » |
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Sir Col
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Re: rank and column space of a matrix
« Reply #2 on: May 27th, 2004, 11:10am » |
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on May 27th, 2004, 9:00am, THUDandBLUNDER wrote:
Your punctuation is rather poor. You should write "Explain!" |
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Your punctuation is almost as poor. You should have written...
Your punctuation is rather poor. You should write, "Explain!" 
Nikki, I'm afraid that I don't understand the question. I thought that the null space is the set of vectors, v, that solve the equation, Av = 0. How can a vector be a line through the origin? Perhaps one of our resident experts in Linear Algebra can assist.
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Icarus
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Re: rank and column space of a matrix
« Reply #3 on: May 27th, 2004, 7:57pm » |
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One vector cannot be a line. However a "set of vectors" can be a line! The nullspace is presumably exactly the set Sir Col identified - though it is more often called the "kernal". This set is easily seen to be closed under addition and scalar multiplication, so it must be a vector space itself. Therefore it must consist of the origin alone, or a line through the origin, or a plane through the origin, or all of space.
In this case, we are given that it is a line, which means that the matrix A has exactly two linearly independent rows or columns.
This is the best I can tell you though, because the terminology "Row or column space" is not one I have ever heard before. What do you mean by these?
As for (b), the equation is: Rank + dimension of kernal (nullspace) = Dimension of the domain.
Since Ax = 0 has a non-trivial solution, what is the smallest dimension possible of the nullspace?
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Eigenray
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Re: rank and column space of a matrix
« Reply #4 on: May 27th, 2004, 11:23pm » |
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The row/column space is the span of the rows/columns. These spaces have the same dimension, which is the rank of the matrix. The dimension of the nullspace is called the nullity.
Nikki, to relate the rank and nullity of a matrix, there is a theorem called, conveniently enough, the "rank-nullity theorem", and for an m x n matrtix, tells you that rank A + nullity A = n.
So, if the nullspace is a line, then its dimension is what? Then what is the rank, i.e., dimension of the row or column space? Can they be lines?
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ThudnBlunder
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Re: rank and column space of a matrix
« Reply #5 on: May 29th, 2004, 3:24am » |
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Quote:| ...though it is more often called the "kernal". |
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...but not outside of Kansas - 'round these here parts it goes by the name of 'kernel'. 
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| « Last Edit: Jun 3rd, 2004, 3:06am by ThudnBlunder » |
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