Question

True or False? No reasons needed.

(e) Suppose β and γ are bases of F n and F m, respectively. Every m × n matrix A is equal to [T] γ β for some linear transformation T: F n → F m.

(f) Recall that P(R) is the vector space of all polynomials with coefficients in R. If a linear transformation T: P(R) → P(R) is one-to-one, then T is also onto.

(g) The vector spaces R 5 and P4(R) are isomorphic.

(h) Let T,U: V → W be linear transformations and β, γ ordered bases of V and W respectively. If [T] γ β = [U] γ β , then T = U.

(i) If A and B are n × n matrices, then AB is invertible if and only if both A and B are invertible.

(j) An n×n matrix A is invertible if and only if its associated linear transformation LA : F n → F n is an isomorphism.

(k) There exists a linear transformation R 2 → R 4 that is onto.

Answer #1

Suppose V is a vector space over F, dim V = n, let T be a linear
transformation on V.
1. If T has an irreducible characterisctic polynomial over F,
prove that {0} and V are the only T-invariant subspaces of V.
2. If the characteristic polynomial of T = g(t) h(t) for some
polynomials g(t) and h(t) of degree < n , prove that V has a
T-invariant subspace W such that 0 < dim W < n

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!

Answer all of the questions true or false:
1.
a) If one row in an echelon form for an augmented matrix is [0 0 5
0 0]
b) A vector b is a linear combination of the columns of a matrix A
if and only if the
equation Ax=b has at least one solution.
c) The solution set of b is the set of all vectors of the form u =
+ p + vh
where vh is any solution...

3. Let V and W be ﬁnite-dimensional vector spaces over ﬁeld F
with dim(V) = n and dim(W) = m, and
let φ : V → W be a linear transformation. Fill in the six blanks
to give bounds on the sizes of the
dimension of ker(φ) and the dimension of im(φ).
3. Let V and W be ﬁnite-dimensional vector spaces over ﬁeld F
with dim(V ) = n and dim(W) = m, and
let φ : V → W...

True or False
A.The dot product of two vectors is always less than the
product of the lengths of the vectors.
B. All linear combinations of the vectors u= (u1,u2) and
v= (v1,v2) form the parallelogram whose adjacent sides are u and
v.
C. If A and B are square matrices of the same size, then
the second column of AB can be obtained by multiplying the second
column of A to the matrix B.
D. The vector that starts...

i) F o r t h e f o l l o w I n g f i n d t h e ( c o m p. E x
p.) f o u r I e r s e r i e s f o r x( t )
I I ) D r a w t h e am p &. P h a s e s p e c t r a
I I I ) T...

1. Let V and W be ﬁnite-dimensional vector spaces over ﬁeld F
with dim(V) = n and dim(W) = m, and
let φ : V → W be a linear transformation.
A) If m = n and ker(φ) = (0), what is im(φ)?
B) If ker(φ) = V, what is im(φ)?
C) If φ is surjective, what is im(φ)?
D) If φ is surjective, what is dim(ker(φ))?
E) If m = n and φ is surjective, what is ker(φ)?
F)...

Determine whether the given set ?S is a subspace of the vector
space ?V.
A. ?=?2V=P2, and ?S is the subset of ?2P2
consisting of all polynomials of the form
?(?)=?2+?.p(x)=x2+c.
B. ?=?5(?)V=C5(I), and ?S is the subset of ?V
consisting of those functions satisfying the differential equation
?(5)=0.y(5)=0.
C. ?V is the vector space of all real-valued
functions defined on the interval [?,?][a,b], and ?S is the subset
of ?V consisting of those functions satisfying
?(?)=?(?).f(a)=f(b).
D. ?=?3(?)V=C3(I), and...

Consider permutations of the 26-character lowercase alphabet
Σ={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}.
In how many of these permutations do
a,b,c occur consecutively and in that
order?
In how many of these permutations does a appear before
b and b appear before c?

MULTIPLE CHOICE. Choose the one alternative that best completes
the statement or answers the question.
6) Consider the following dataset:
ID
V
W
X
Y
Z
1
10
2
8
Group 1
A
2
2
8
3
Group 1
A
3
2
8
5
Group 2
A
4
2
10
2
Group 2
A
5
5
2
5
Group 2
A
6
5
5
3
Group 1
B
7
10
7
1
Group 2
B
8
2
9
10
Group...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 7 minutes ago

asked 9 minutes ago

asked 11 minutes ago

asked 11 minutes ago

asked 19 minutes ago

asked 41 minutes ago

asked 43 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago