3. (a) (2 marks) Consider R 3 over R. Show that the vectors (1,
2, 3)...
3. (a) Consider R 3 over R. Show that the vectors (1,
2, 3) and (3, 2, 1) are linearly independent. Explain why they do
not form a basis for R 3 .
(b) Consider R 2 over R. Show that the vectors (1, 2),
(1, 3) and (1, 4) span R 2 . Explain why they do not form a basis
for R 2 .
Consider the linearly independent set of vectors
B= (-1+2x+3x^2+4x^3+5x^4, 1-2x+3x^2+4x^3+5x^4,
1+2x-3x^2+4x^3+5x^4, 1+2x+3x^2-4x^3+5x^4, 1+2x+3x^3+4x^3-5x^4)
in P4(R), does...
Consider the linearly independent set of vectors
B= (-1+2x+3x^2+4x^3+5x^4, 1-2x+3x^2+4x^3+5x^4,
1+2x-3x^2+4x^3+5x^4, 1+2x+3x^2-4x^3+5x^4, 1+2x+3x^3+4x^3-5x^4)
in P4(R), does B form a basis for P4(R) and why?
How many trees T are there on the set of vertices
{1, 2, 3, 4, 5,...
How many trees T are there on the set of vertices
{1, 2, 3, 4, 5, 6, 7} in which the vertices 2 and 3 have degree 3,
vertex
5 has degree 2, and hence all others have degree 1? Do not
just
draw pictures but consider the possible Pr¨ufer codes of these
trees.
a. Find an orthonormal basis for R 3 containing the vector (2,
2, 1). b. Let...
a. Find an orthonormal basis for R 3 containing the vector (2,
2, 1). b. Let V be a 3-dimensional inner product space and S = {v1,
v2, v3} be an orthonormal set in V. Explain whether the set S can
be a basis for V .
1. Let T = {(1, 2), (1, 3), (2, 5), (3, 6), (4, 7)}. T :...
1. Let T = {(1, 2), (1, 3), (2, 5), (3, 6), (4, 7)}. T : X ->
Y. X = {1, 2, 3, 4}, Y = {1, 2, 3, 4, 5, 6, 7}
a) Explain why T is or is not a function.
b) What is the domain of T?
c) What is the range of T?
d) Explain why T is or is not one-to one?
(a) Do the vectors v1 = 1 2 3 , v2 = √ 3 √ 3...
(a) Do the vectors v1 = 1 2 3 , v2 = √ 3 √ 3 √ 3 , v3=√ 3 √ 5 √
7, v4 = 1 0 0 form a basis for R 3 ? Why or why not?
(b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and
a2, where a1 = (1 0 −1 0) , a2 = 0 1 0 −1. Find a basis for the
orthogonal complement V ⊥...
Do the vectors v1 = 1 2 3 ,
v2 = ...
Do the vectors v1 = 1 2 3 ,
v2 = √ 3 √ 3 √ 3 ,
v3 √ 3 √ 5 √ 7 ,
v4 = 1 0 0 form a basis for R 3 ? Why or why not?
(b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and
a2, where a1 = ...