In each of the attached cases show that the set U is linearly independent subset of the vectorspace V. Give, with justification, a basis of V which contains the set U.

Solution Summary

This explains how to show a subset is linearly independent subset of the vector space V.

Consider R2 with the following rules of multiplications and additions: For each x=(x1,x2), y=(y1,y2):
x+y=(x2+y2,x1+y1) and for any scalar alpha, alpha*x=(alpha*x1, alpha*x2)
Is it a vector space, if not demonstrate which axioms fail to hold. Also, show that Pn- the space of polynomials of order less than n is a vector spac

Define vectors pace and subspace with examples.
State and prove a necessary and sufficient condition for a subset of vectors to be a subspace.
Show that the intersection and union of two sub spaces are also sub spaces.

How to prove or counter with example the following statements:
(1) If two subspaces are orthogonal, then they are independent.
(2) If two subspaces are independent, then they are orthogonal.
I know that a vector v element of V is orthogonal to a subspace W element V if v is orthogonal to every w element W. Two subspaces W1

1) Show that if dim X = 1 and T belongs to L(X,X), there exists k in K st Tx=kx for all x in X.
2) Let U and V be finite dimensional linear spaces and S belong to L(V,W), T belong to L(U,V). Show that the dimension of the null space of ST is less than or equal to the sum of the dimensions of the null spaces of S and T.
3)

Let F be the field of real numbers and let V be the set of all sequences:
(a_1, a_2, ..., a_n, ...), a_i belongs to F, where equality, addition and scalar multiplication are defined component wise. Then V is a vector space over F.
Let W = {(a_1, a_2, ..., a_n, ...) belongs to V | lim n -> infinity a_n = 0}.
Prove t

2. Which of the following sets of vectors in R3 are linearly dependent?
a) (4, -1, 2), (-4, 10, 2)
b) (-3, 0, 4), (5, -1, 2), (1, 1, 3)
c) (8, -1, 3), (4, 0, 1)
d) (-2, 0, 1), (3, 2, 5), (6, -1, 1), (7, 0, -2)
4. Which of the following sets of vectors in P2 are linearly dependent?
a) 2 - x + 4x2, 3 + 6x + 2x2, 2

Give a demonstration as to why or why not the given objects are vector subspaces of M22
a) all 2 X 2 matrices with integer entries
A vector space is a set that is closed under finite vector addition and scalar multiplication.
It is not a vector space, since V is NOT closed under finite scalar multiplication. For insta

The medial triangle of a triangle ABC is the triangle whose vertices are located at the midpoints of the sides AB, AC, and BC of triangle ABC. From an arbitrary point O that is not a vertex of triangle ABC, you may take it as a given fact that the location of the centroid of triangle ABC is the vector (vector OA + vector OB + ve