In: Math

Let V be the vector space of 2 × 2 real matrices and let P2 be the vector space of polynomials of degree less than or equal to 2. Write down a linear transformation T : V ? P2 with rank 2. You do not need to prove that the function you write down is a linear transformation, but you may want to check this yourself. You do, however, need to prove that your transformation has rank 2.

Let A =

a |
b |

c |
d |

be an arbitrary 2x2 matrix in V where a, b, c, d are arbitrary
real numbers. Also, let p(x) = ax^{2}+bx be an arbitrary
polynomial in P_{2}. Now, let us define T : V ?
P_{2} by T(A) = p(x).

Further, let
{E_{11},E_{12},E_{21},E_{22}}
denote the standard basis of V where E_{ij} is the 2x2
matrix with 1 as the ijth entry, the remaining entries being 0.
Also,{ 1,x,x^{2}} is the standard basis of
P_{2}.

Now, T(E_{11})=
x^{2},T(E_{12})=x,T(E_{21})=0 and
T(E_{22})=0. Then the standard matrix of T is
[T(E_{11}),T(E_{12}),T(E_{21}),T(E_{22})]
= M =

1 |
0 |
0 |
0 |

0 |
1 |
0 |
0 |

0 |
0 |
0 |
0 |

It may be observed that the entries in the columns of M are the
coefficients of x^{2} and x in T(E_{11}),
T(E_{12}), T(E_{21}) and T(E_{22})
respectively.

It may also be observed that M has 2 non-zero rows so that rank(M) = 2. Therefore, rank(T) = rank(M) = 2.

Let A =

a |
b |

c |
d |

and B =

e |
f |

g |
h |

be 2 arbitrary matrices in V and let k be an arbitrary scalar. Then A+B =

a+e |
b+f |

c+g |
d+h |

and kA =

ka |
kb |

kc |
kd |

so that T(A+B) = (a+e)x^{2}+(b+f)x = ax^{2}+bx+
ex^{2}+fx = T(A) +T(B). Thus, T preserves vector
addition.

Also, T(kA) = kax^{2}+kbx = k(ax^{2}+bx) =
kT(A). Thus, T preserves scalar multiplication. Hence T is a linear
transformation.

Let V = R^2×2 be the vector space of 2-by-2 matrices with real
entries over
the scalar field R. We can define a function L on V by
L : V is sent to V
L = A maps to A^T ,
so that L is the “transpose operator.” The inner product of two
matrices B in R^n×n and C in R^n×n is usually defined to be
<B,C> := trace (BC^T) ,
and we will use this as our inner...

Let V be a vector space and let U and W be subspaces of V . Show
that the sum U + W = {u + w : u ∈ U and w ∈ W} is a subspace of V
.

1. Let V be real vector space (possibly infinite-dimensional),
S, T ∈ L(V ), and S be in- vertible. Prove λ ∈ C is an eigenvalue
of T if and only if λ is an eigenvalue of STS−1. Give a description
of the set of eigenvectors of STS−1 associated to an eigenvalue λ
in terms of the eigenvectors of T associated to λ.
Show that there exist square matrices A, B that have the same
eigenvalues, but aren’t similar. Hint:...

6. Let V be the vector space above. Consider the maps T : V → V
And S : V → V
defined by T(a1,a2,a3,...) = (a2,a3,a4,...) and S(a1,a2,a3,...)
= (0,a1,a2,...).
(a) [optional] Show that T and S are linear.
(b) Show that T is surjective but not injective.
(c) Show that S is injective but not surjective.
(d) Show that V = im(T) + ker(T) but im(T) ∩ ker(T) ̸= {0}.
(e) Show that im(S) ∩ ker(S) = {0}...

Let P2 be the vector space of all polynomials of
degree less than or equal to 2.
(i) Show that {x + 1, x2 + x, x − 1} is a basis for
P2.
(ii) Define a transformation L from P2 into
P2 by: L(f) = (xf)' . In other words,
L acts on the polynomial f(x) by first multiplying the function by
x, then differentiating. The result is another polynomial in
P2. Prove that L is a linear transformation....

Proof:
Let S ⊆ V be a subset of a vector space V over F. We have that S
is linearly dependent if and only if there exist vectors v1, v2, .
. . , vn ∈ S such that vi is a linear combination of v1, v2, . . .
, vi−1, vi+1, . . . , vn for some 1 ≤ i ≤ n.

(10pt) Let V and W be a vector space over R. Show that V × W
together with (v0,w0)+(v1,w1)=(v0 +v1,w0 +w1) for v0,v1 ∈V, w0,w1
∈W
and
λ·(v,w)=(λ·v,λ·w) for λ∈R, v∈V, w∈W is a vector space over
R.
(5pt)LetV beavectorspaceoverR,λ,μ∈R,andu,v∈V. Provethat
(λ+μ)(u+v) = ((λu+λv)+μu)+μv.
(In your proof, carefully refer which axioms of a vector space
you use for every equality. Use brackets and refer to Axiom 2 if
and when you change them.)

Let U be a subset of a vector space V. Show that spanU is the
intersection of all the subspaces of V that contain U. What does
this say if U=∅? Need proof

Let V be a vector space, and suppose that U and W are both
subspaces of V. Show that U ∩W := {v | v ∈ U and v ∈ W} is a
subspace of V.

Let V be a finite-dimensional vector space over C and T in L(V).
Prove that the set of zeros of the minimal polynomial of T is
exactly the same as the set of the eigenvalues of T.

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