Question

Let V be the set of all triples (r,s,t) of real numbers with the standard vector addition, and with scalar multiplication in V deﬁned by k(r,s,t) = (kr,ks,t). Show that V is not a vector space, by considering an axiom that involves scalar multiplication. If your argument involves showing that a certain axiom does not hold, support your argument by giving an example that involves speciﬁc numbers. Your answer must be well-written.

Answer #1

Let V be the set of all ordered pairs of real numbers. Consider
the following addition and scalar multiplication operations V. Let
u = (u1, u2) and v = (v1, v2).
• u ⊕ v = (u1 + v1 + 1, u2 + v2 + )
• ku = (ku1 + k − 1, ku2 + k − 1)
Show that V is not a vector space.

Let S={(0,1),(1,1),(3,-2)} ⊂ R², where R² is a real vector space
with the usual vector addition and scalar multiplication.
(i) Show that S is a spanning set for R²
(ii)Determine whether or not S is a linearly independent set

Let V be the set of all ordered pairs of real numbers. Consider
the following addition and scalar multiplication operations V. Let
u = (u1, u2) and v = (v1, v2).
• u ⊕ v = (u1 + v1 + 1, u2 + v2 + )
• ku = (ku1 + k − 1, ku2 + k − 1)
1)Show that the zero vector is 0 = (−1, −1).
2)Find the additive inverse −u for u = (u1, u2). Note:...

Let P4 denote the space of polynomials of degree less than 4
with real coefficients. Show that the standard operations of
addition of polynomials, and multiplication of polynomials by a
scalar, give P4 the structure of a vector space (over the real
numbers R). Your answer should include verification of each of the
eight vector space axioms (you may assume the two closure axioms
hold for this problem).

Consider the set of all ordered pairs of real numbers with
standard vector addition but with scalar multiplication defined
by k(x,y)=(k^2x,k^2y).
I know this violates (alpha + beta)x = alphax + betax, but I'm
not for sure how to figure that out? How would I figure out which
axioms it violates?

Are the following vector space and why?
1.The set V of all ordered pairs (x, y) with the addition of
R2, but scalar multiplication a(x, y) = (x, y) for all a
in R.
2. The set V of all 2 × 2 matrices whose entries sum to 0;
operations of M22.

Let V = Pn(R), the vector space of all polynomials of degree at
most n. And let T : V → V be a linear transformation. Prove that
there exists a non-zero linear transformation S : V → V such that T
◦ S = 0 (that is, T(S(v)) = 0 for all v ∈ V) if and only if there
exists a non-zero vector v ∈ V such that T(v) = 0.
Hint: For the backwards direction, consider building...

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