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Is thi set of vectors, $\ { (2, 1), (3, 2), (1, 2)\}$, is linearly . . . When you have vectors like $\displaystyle\left [\begin {array} {c}1\\2\end {array}\right]=1\mathbf {i}+2\mathbf {j}$, they live on the plane where there are essentially two different directions: left right ($\mathbf {i}$) and up down ($\mathbf {j}$) Every other direction can be made out of combinations of left right and up down; e g $1\mathbf {i}+2\mathbf {j}$ Now for a set to be linearly
How do the definitions of irreducible and prime elements differ? You are correct in observing that your lecture notes are not quite right here It is indeed the definition of an irreducible element What you read elsewhere is indeed the definition of prime element
What does versus mean in the context of a graph? The answer (as is often the case) come from Latin "versus" simply means against and is used in the sporting context as well We say that in some contest "Team A versus team B", meaning team A is against team B The graph is the same - one variable is plotted against (or versus) another From the same cognate root we also get the English "adversary"
linear algebra - Determine if vectors are linearly independent . . . Firstly, you are to arrange the vectors in a matrix form the reduce them to a row-reduced echelon form (If the last row becomes all zeros then it is linearly dependent, but if the last row isn't all zeros then it is linearly independent) Let's get to it now Arranging the vectors in matrix form we have : \begin {bmatrix}2 2 0 \\ 1 -1 1 \\ 4 2 -2 \end {bmatrix} After the first
The variance of Brownian motion - Mathematics Stack Exchange The notation on the second section of quotation and in your block of questions is not quite consistent, but the underlying answer is yes For a standard Wiener process, as stated in the first section of quotation, the change in a period of time $\Delta t$ is a random variable normally distributed with mean $0$ and variance $\Delta t$, i e standard deviation $\sqrt {\Delta t}$, so is equal to