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$\begingroup$

If my reduced row echelon of the matrix looks like

\begin{pmatrix} 1 & 0 & 0 & 0 & 0 &0\\ 0 & 1 & 0 & 0 &0 &0\\ 0 & 0 & 0 & 0 &0 & 0\\ 0 & 0 & 0 & 0 &0 & 0\\ 0 & 0 & 0 & 0 &0 &0\end{pmatrix}

then using the linear algebra terminology I would say that the dimension or the degrees of freedom is two correct?

What is meant by $Ker(M) \neq 0$? I don't understand this and would need an example. Also if the $Ker(M) = 0$ is the matrix $M$ invertible? Does the converse of this statement hold? Please help I want to get this material to pass my linear algebra class

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    Dimension of *what* is two? The dimension of the row space is two; the dimension of the column space is two; the rank is two. The kernel is the nullspace. the zero-vector is always in the nullspace. Kernel not zero means the zero vector isn't the only element of the nullspace. If the kernel is zero, *and the matrix is square*, then, yes, it is invertible.2012-10-27
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    The dimension of the matrix that I put up.2012-10-27
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    A matrix doesn't have a dimension. If you like, it has *two* dimensions, a height and a length --- your matrix has dimensions $5\times6$.2012-10-27
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    @depressed Dimension has a very specialized meaning in linear algebra. By "degrees of freedom" do you mean the number of _free variables_?2012-10-27
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    Could you give me an example of a matrix $M$ where the kernel is not zero? And if the kernel is zero how many vectors are there in the nullspace?2012-10-27
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    I have written down in my notes that the "dimension of a space is the number of degrees of freedom of the space."2012-10-27
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    The matrix you have given has kernel not zero. Do you know what the nullspace of a matrix is?2012-10-27
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    @depressed: That is an extremely vague statement. Did your instructor talk about vector spaces at all?2012-10-27
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    By kernel being zero, we really mean that $\ker(M) = \{\mathbf{0}\}$, so by definition, if the kernel is zero (or more technically _trivial_) then it contains only the zero vector. And that is a horrible definition of dimension.2012-10-27
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    @depressed: Do you realize that nullspace is another name for kernel?2012-10-27
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    @wj32, yes I do know that the nullspace is another name for kernel. I had to study that for a midterm. The professor asked us to study the terms.2012-10-27
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    @depressed Can you outline briefly what exactly you have learned so far? It doesn't have to be very detailed, just the general ideas so we know how to approach your problem.2012-10-27
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    @EuYu: We have studied the kernel (nullspace), dimension, rank, span, eigenvalues, basis, gaussian elimination and discussed something about free variables.2012-10-27
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    @depressed: Can you tell us what a basis is, according to your instructor?2012-10-27
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    A basis is a set of linearly independent vectors that can form any vector in the vector space roughly speaking.2012-10-27
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    @ Gerry: What other vector is in the nullspace of the matrix besides the zero vector?2012-10-27
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    $\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}^\mathrm{T}$ for example.2012-10-27
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    $(0 0 0 1 0 0)^T$ is another. Thanks.2012-10-27

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