1) Suppose v1, v2, V3, V4 span a vector space V. Prove that the list V1-V2, V2 – V3, V3 – V4, V4 also spans V. [31 2 F5T 9 is not linearly independent in R3. 5. 2) Find a number t such that 1,-3 [4] 3) (a) Show that if we consider C a vector space orver R, then the list {1+ i,1 - i} is linearly independent. (b) Show that if we consider Ca vector space orver C, then the list {1 + i, 1- i} is inearly dependent. 4) Prove or give a counterexample: If v1, v2,..Vm is a linearly independent list of vectors in a vector space V (over either Q, R, C), then 5v, - 4vz, Vz, ... Vm is linearly independent. 5) Prove or give a counterexample: If v1, Vz, ...Vm and W1, W2, ... Wm are linearly independent lists of vectors in a vector space V (over either Q, R, C), then 5v1 + W1, V2 + W2, .. Vm + Wm is linearly independent. 6) Let u, v be vectors in the space V pver the field F and c a scalar. Prove that Span(u, v) = Span(u, cu + v). 7) Let u, v be vectors in the space V pver the field F and ca non-zero scalar. Prove that Span(u, v) = Span(cu, v) 8) Let u, v be non-zero vectors. Prove that (u, v) is linearly dependendent if and only the vectors are scalar muitiplies of one another. 9) Let V be a vector space and assume that (v1, v2, V3) is a linearly independent sequence from V, w is a vector from V, and that (v1 + w, v2 + w, v3 + w) is a linearly dependent. Prove that w E Span(v1, v2, v3).
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
Question 2
![1) Suppose v1, V2, V3, V4 span a vector space V. Prove that the list
V1-V2, V2 - V3, V3 – V4, V4 also spans V.
[3] [ 21 [51
2) Find a number t such that 1,-3,19 is not linearly independent in R³.
3) (a) Show that if we consider C a vector space orver R, then the list {1+ i,1- i} is
linearly independent.
(b) Show that if we consider Ca vector space orver C, then the list {1+i, 1- i} is
linearly dependent.
4) Prove or give a counterexample: If v1, v2,... Vm is a linearly independent list of
vectors in a vector space V (over either Q, R, C), then 5v, - 4v2, V2, .. Vm
linearly independent.
5) Prove or give a counterexample: If v, vz,.. Vm and w1, W2, ... Wm are inearly
independent lists of vectors in a vector space V (over either Q, R, C), then 5v1 +
W1, V2 + W2, . Vm + Wm is linearly independent.
6) Let u, v be vectors in the space V pver the field IF and c a scalar. Prove that
Span(u, v) = Span(u, cu + v).
7) Let u, v be vectors in the space V pver the field IF and c a non-zero scalar. Prove
that Span(u, v) = Span(cu, v)
8) Let u, v be non-zero vectors. Prove that (u, v) is linearly dependendent if and
only the vectors are scalar multiplies of one another.
9) Let V be a vector space and assume that (v1, v2, V3) is a linearly independent
sequence from V, w is a vector from V, and that (v1 + w, vz + w, v3+ w) is a
linearly dependent. Prove that w e Span(v1, v2, V3).
is
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