Midterm Study Cards Flashcards
What is a linear combination?
The linear combination of two vectors v and w is av + bw where a and b are scalars
Are Point and Vector equivalent in linear algebra?
Yes
What is the symbol for and the formula for the length/magnitude of a n-vector v
The symbol and equation is
||v||, sqrt((v_1)^2+(v_2)^2+(v_n)^2)
How do you prove two vectors are linearly independent?
If there is no way to form a linear combination of them equal to 0 without a and b being 0
What is the equation of the distance between two vectors v and w?
The equation is
||v - w||
What is the zero vector?
The zero vector in R^n with coordinates [0,0,0,0,…,0]
What is a unit vector?
A unit vector is a vector with length 1
When are vectors x and y perpendicular
When the dot product of x and y is 0
What is the cosine of the angle between two vectors v and w?
The cosine is
(v dot w)/(||v||*||w||)
When are two vectors orthogonal?
When they are perpendicular to each other
What 5 rules hold up for all vectors?
v dot w = w dot v
v dot v = ||v||^2
v dot (cw) = c(v dot w) (where c is a scalar)
v dot (w1 + w2) = (v dot w1) + (v dot w2)
v dot (c1w1 + c2w2) = c1(v dot w1) + c2(v dot w2)
What is the correlation coefficient between two vectors v and w?
The equation is
(v dot w)/(||v||*||w||) where v_avg and w_avg = 0
What does a correlation coefficient close to -1 imply?
The data points are close to a line of negative slope
What does a correlation coefficient close to 1 imply?
The data points are close to a line with positive slope
What does a correlation coefficient close to 0 imply?
x and y do not have a strong linear correlation
What is the basic equation of a plane?
ax + by + cz = d where a, b, c, and d are constants
What is the parametric form of the equation of a plane?
P + te + t’e’ where t and t’ are all possible scalar values
What is the normal vector form of the equation of a plane?
A plane is defined by stating it is perpendicular to the vector n and passes through the point w.