Linear Algebra 2) General Inner Products, Length, Distance and Angle a) Determine if (u,v)-3uiv,-u,v, is a dot product b) Show that (u.v)-a+a,h,'2 is a product if a, 20 e)Let A-(41 ..)and B-G ) Use inner product on 4 -2 M (A, B aitai +apb +2a to find the length of A, B, namely ll-41 and 1 d) Find the angle between the two matrices above e) Find the distance between the two above matrices 0) For the functions (x)-1 and...
Compute the dot product u v. 2 U.V = -3+ 16-24 -1| 3 10) u = For the given matrix A, find a basis for the corresponding eigenspace for the given eigenvalue - 16 - 16] 16, A= -7 11) A = 0 0-8 - 15 16
Properties of the dot product Please help! theoretical calculus 2. Some properties of the dot product: (a) The Cauchy-Schwartz inequality: Given vectors u and v, show that lu-vl lullv1. When is this inequality an equality? (Hint: Use the relationship between u-v and the angle θ between u and v.) (b) The dot product is positive definite: Show that u u 2 0 for any vector u and that u u 0 only when u-0. (c) Find examples of vectors u,...
Let u(t)= ti+ln(t)j + et k and v(t) = ti+2tj+1k. Compute the derivative of the dot product [u(t)- v(t)] in two ways and confirm they agree: • Compute the dot product u(t). v(t) first and then differentiate the result. • Alternatively, use the following “Dot Product Rule" v(t)] = u'(t) . v(t) + u(t) . v'(t). (1)
Let u and v be column vectors containing real numbers. Describe how to calculate the dot product of u and v. What is the difference between a discrete random variable and a continuous random variable? data science
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0}, where w.v is the dot product. 2 (a) Prove that U is a subspace of R3. (b) Find a basis for U and compute its dimension. 4. Decide whether or not the following subsets of vector spaces are linearly independent. If they are, prove it. If they aren't, write one as a linear combination of the others. (a) The subset {0 0 0 of...
4. Consider R³ with the standard inner product (i.e. the dot product). Let c{() (:} U = span 2 Find U+. (Write as a span of some set.)
Let V be a real inner product space. Under what condition on u, v E V is the following equality valid: where l 1x12 = (x, x) YE V.
Given the vector U=(4,3) and V=(1,-1) (a) Write U in terms of I, J (b) Find the exact magnitude of U (c) Find U+2V (d) Find the dot product, UV (e) Are U and V perpendicular? Explain in terms of the dot product