Question

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

Answer #1

Consider C as a vector space over R in the natural way. Here
vector addition and scalar
multiplication are the usual addition and multiplication of
complex numbers. Show that {1 − i, 1 + i} is
linearly independent. Consider C as a vector space over C in
the natural way. Here vector addition is the
usual addition of complex numbers and the scalar
multiplication is the usual multiplication of a real number
by a complex number. Show that {1 −...

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.

Consider the vector space M2x2 with the usual addition and
scalar multiplication. Let it be the subspace of M2x2 defined as
follows:
H= { | a b |
| c d | with b= c}
consider matrices A= | 1 2 | B= |-2 2 |
C= | 1 8 |
| 1 3 | , | 1 -3 | , | 4 6 |
Do they form a basis for H? justify the answer

Determine whether the set with the definition of addition of
vectors and scalar multiplication is a vector space. If it is,
demonstrate algebraically that it satisfies the 8 vector axioms. If
it's not, identify and show algebraically every axioms which is
violated. Assume the usual addition and scalar multiplication if
it's not defined. V = R, x + y = max( x , y ), cx=(c)(x) (usual
multiplication.

Suppose we have a vector space V of dimension n. Let R be a
linearly independent set with order n−2. Let S be a spanning set
with order n+ 2. Outline a strategy to extend R to a basis for V.
Outline a strategy to pare down S to a basis for V .

Let V be a vector space: d) Suppose that V is
finite-dimensional, and let S be a set of inner products on V that
is (when viewed as a subset of B(V)) linearly independent. Prove
that S must be finite
e) Exhibit an infinite linearly independent set of inner
products on R(x), the vector space of all polynomials with real
coefficients.

use the subspace theorem ( i) is it a non-empty space? ii) is it
closed under vector addition? iii)is it closed under scalar
multiplication?) to decide whether the following is a real vector
space with its usual operations:
the set of all real polonomials of degree exactly n.

Determine whether the set with the definition of addition of
vectors and scalar multiplication is a vector space. If it is,
demonstrate algebraically that it satisfies the 8 vector axioms. If
it's not, identify and show algebraically every axioms which is
violated. Assume the usual addition and scalar multiplication if
it's not defined. V = R^2 , < X1 , X2 > + < Y1 , Y2 > =
< X1 + X2 , Y1 +Y2> c< X1 , X2...

Determine if W is a subspace of R^3 under the usual addition and
scalar multiplication. Either show algebraically that it is or show
how it isn't algebraically. W= {(x1, x2, x3) ∈ R^3 x1 = x2 and x2 =
2x3 }

Show that the set GLm,n(R) of all mxn matrices with
the usual matrix addition and scalar multiplication is a finite
dimensional vector space
with dim GLm,n(R) = mn.
Show that if V and W be finite dimensional vector spaces with
dim V = m and dim W = n, B a basis for V and C a basis for W
then
hom(V,W)-----MatB--->C(-)-------->
GLm,n(R) is a bijective linear transformation. Hence or
otherwise, obtain dim hom(V,W).
Thank you!

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