Do you mean not necessarily independent? As Matt Grime pointed out, if the dimension of the space is less than 3, it is true that no set of 3 vectors is independent- the condition that any pair are independent is irrelevant while if the dimension is three or greater, this is not necessarily true. In three or more dimensions it is the case that if a set of 3 vectors is independent, then any pair are independent.physicsss said:prove if {v1,v3}, {v2,v3}, and {v1,v3} are all linearly independent, then{v1,v2,v3} is not linearly independent.
I'm having trouble showing that is true other than showing a counter example when it doesn't work, namely when v1=1,v2=0, and v3=1.
TIA.
Independent variables are variables that are manipulated or controlled by the researcher, while dependent variables are the outcome or result of the manipulation of the independent variable.
The best way to determine whether a variable is independent or dependent is to ask yourself: "Is this variable being manipulated or controlled by the researcher?" If the answer is yes, then it is an independent variable. If the answer is no, then it is a dependent variable.
No, a variable cannot be both independent and dependent at the same time. However, a variable can be independent in one study and dependent in another study, depending on how it is being manipulated.
Distinguishing between independent and dependent variables is important because it helps to establish cause and effect relationships in research. By manipulating the independent variable and observing the changes in the dependent variable, researchers can determine whether the independent variable has a direct impact on the dependent variable.
Confounding variables are variables that can affect the relationship between the independent and dependent variables. To control for confounding variables, researchers must carefully design their study and use techniques such as randomization, control groups, and statistical analysis to minimize their impact on the results.