are linearly independent if and only if?
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A set of two vectors is linearly independent if and only if neither of the vectors is a multiple of the other. A set of vectors S = {v1,v2,...,vp} in Rn containing the zero vector is linearly dependent. Theorem If a set contains more vectors than there are entries in each vector, then the set is linearly dependent.
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Indeed x/x is only defined when x≠0. And wherever it is defined, its value is 1.
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Linearly dependent means “yes, you can”, linearly independent means, “no, you can't”. ... Linearly dependent means “yes, you can”, linearly independent means, “no, you can't”. So for example, a single vector being linearly dependent means that you can multiply it by a non-zero scalar and get the zero vector.
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Product rule for independent events If A and B are independent, P(AB)=P(A)P(B) (because P(A|B)=P(A) for independent events). (Example: If A and B are independent and P(A)=.
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Consider two discrete random variables X and Y. ... Intuitively, two random variables X and Y are independent if knowing the value of one of them does not change the probabilities for the other one. In other words, if X and Y are independent, we can write P(Y=y|X=x)=P(Y=y), for all x,y.
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IF AND ONLY IF, is a biconditional statement, meaning that either both statements are true or both are false. So it is essentially and “IF” statement that works both ways. Note that IF AND ONLY IF is different than simply ONLY IF.
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(1) T is one-to-one if and only if the columns of A are linearly independent, which happens precisely when A has a pivot position in every column. (2) T is onto if and only if the span of the columns of A is Rm, which happens precisely when A has a pivot position in every row.
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Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent, P(A and B) ... For any two events A and B, P(A or B) = P(A) + P(B) - P(A and B).
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Suppose two events have a non-zero chance of occurring. Then if the two events are mutually exclusive, they can not be independent. If two events are independent, they cannot be mutually exclusive.
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System of rows of square matrix are linearly independent if and only if the determinant of the matrix is ​​not equal to zero. Note. System of rows of square matrix are linearly dependent if and only if the determinant of the matrix is equals to zero. Example 1.
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1 Expert Answer This argument shows that if two events are independent, then each event is independent of the complement of the other. Therefore A' and B' are also independent events.
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If Wronskian W(f,g)(t0) is nonzero for some t0 in [a,b] then f and g are linearly independent on [a,b]. If f and g are linearly dependent then the Wronskian is zero for all t in [a,b]. Show that the functions f(t) = t and g(t) = e2t are linearly independent.
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How to use "only if" in English Grammar. 'Only if' means 'only one condition happens". This form basically means the same as 'if'. The main emphasis is on the result. You can go out only if you do your homework.
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The one sentence summary: If A and B are mutually exclusive events, then they are independent if and only if P(A) = 0 or P(B) = 0.
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(1) Only something is used in independent positive statements. We can say Something is wrong, but we can't say *Anything is wrong.
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Absolute Value Function Family If x = 0, then y = |0| = 0. If x is positive, then the function value is equal to x. ... Every absolute value graph will take this shape, as long as the expression inside the absolute value is linear.
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Yes, two events with nonzero probabilities can be both independent and mutually exclusive when their probabilities are equal. ... No, two events with nonzero probabilities cannot be independent and mutually exclusive because independence is the complement of being mutually exclusive.