s =  1 Considerthefollowingvectorsof R 1357  , t =  − 75 − 31  , u =  − 3015  , v =  (over) − 12 13  , a =  1111 x ∈ V satisfying h{ x }i ∩ U 6 = ∅ .  h U i .(2)Findall h U i ,i.e.ﬁndasystemofequationsofde

シェア "s =  1 Considerthefollowingvectorsof R 1357  , t =  − 75 − 31  , u =  − 3015  , v =  (over) − 12 13  , a =  1111 x ∈ V satisfying h{ x }i ∩ U 6 = ∅ .  h U i .(2)Findall h U i ,i.e.ﬁndasystemofequationsofde"

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(1)

Graduate School Entrance Exam 2013 1

1 Consider the following vectors of R ⁴

s =





 1 3 5 7





 , t =







−3 1

−7 5







, u =







−3 0 1 5







, v =





 1 2

−1 3







, a =





 1 1 1 1







, b =





 1

−1 1

−1







Define subsets U and V of R ⁴ by

U = a + h{ s , t }i span , V = b + h{ u , v }i span

Here hSi span denotes the subvectorspace generated by S for any subet S of R ⁴ , and for a vector c ∈ R ⁴ , and a subvectorspace W ⊂ R ⁴ , c + W = { c + w | w ∈ W }.

(1) Find equations defining hUi span , i.e. find a system of equations of degree one whose solution set is hUi span .

(2) Find all x ∈ V satisfying h{ x }i span ∩ U 6= ∅.

?/2/2013 (over)

(2)

Graduate School Entrance Exam 2013 2

2 _Let _h·, _·i be the canonical Euclidean inner product of R ³ , and v ₁ , v ₂ , v ₃ be a normal orthogonal basis with respect to this inner product. Define the linear map f : R ³ → R ³ by

f( x ) = x − h x , v ₂ i v ₁ − h x , v ₁ i v ₂ . Answer the following questions:

(1) Find the representing matrix of f with respect to the basis { v ₁ , v ₂ , v ₃ }.

(2) Find the eigenvalues and eigenvectors of the linear map f .

(3) Show that the representing matrix A of the linear map f with respect to the canonical basis { e ₁ , e ₂ , e ₃ } is symmetric.

(4) Find the eigenvalues and eigenvectors of A.

?/2/2013 (over)

(3)

Graduate School Entrance Exam 2013 3

3 Answer the following questions.

(1) Find the limit

x→0 lim (cos x) ^1/x

.

(2) Let D = {(x, y)|x ≥ 0, y ≥ 0, x ³ + y ³ ≤ 1}. Find the value of the integral Z Z

D

x ⁸ y ⁵ dxdy.

?/2/2013 (over)

(4)

Graduate School Entrance Exam 2013 4

4 (1) Express the Taylor expansion of the function f ( x, y ) = log( x ² + y ² ) at x = y = 1 in the form

f(x, y) = f (1, 1) + f _x (1, 1)(x − 1) + f _y (1, 1)(y − 1) + R 2 (x, y) (Express the error term R ₂ (x, y) in a suitable form).

(2) Draw the shape of the surface z = log(x ² + y ² ).

(3) Let x = y = 1. Determine one effective digit of the change of z = log( x ² + y ² ) if x increases by 0 . 003 and y decreases by 0 . 002. Also, explain the estimate of the error term on which your calculation is based.

?/2/2013 (end)

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Graduate School Entrance Exam 2013 1

1 Consider the following vectors of R 4

s =









 1 3 5 7









 , t =











−3 1

−7 5











, u =











−3 0 1 5











, v =









 1 2

−1 3











, a =









 1 1 1 1











, b =









 1

−1 1

−1











Define subsets U and V of R 4 by

U = a + h{ s , t }i span , V = b + h{ u , v }i span

Here hSi span denotes the subvectorspace generated by S for any subet S of R 4 , and for a vector c ∈ R 4 , and a subvectorspace W ⊂ R 4 , c + W = { c + w | w ∈ W }.

(1) Find equations defining hUi span , i.e. find a system of equations of degree one whose solution set is hUi span .

(2) Find all x ∈ V satisfying h{ x }i span ∩ U 6= ∅.

?/2/2013 (over)

Graduate School Entrance Exam 2013 2

1 Consider the following vectors of R ⁴

Define subsets U and V of R ⁴ by

Here hSi span denotes the subvectorspace generated by S for any subet S of R ⁴ , and for a vector c ∈ R ⁴ , and a subvectorspace W ⊂ R ⁴ , c + W = { c + w | w ∈ W }.

2 _Let _h·, _·i be the canonical Euclidean inner product of R ³ , and v ₁ , v ₂ , v ₃ be a normal orthogonal basis with respect to this inner product. Define the linear map f : R ³ → R ³ by

f( x ) = x − h x , v ₂ i v ₁ − h x , v ₁ i v ₂ . Answer the following questions:

(1) Find the representing matrix of f with respect to the basis { v ₁ , v ₂ , v ₃ }.

(3) Show that the representing matrix A of the linear map f with respect to the canonical basis { e ₁ , e ₂ , e ₃ } is symmetric.

x→0 lim (cos x) ^1/x

(2) Let D = {(x, y)|x ≥ 0, y ≥ 0, x ³ + y ³ ≤ 1}. Find the value of the integral Z Z

x ⁸ y ⁵ dxdy.

4 (1) Express the Taylor expansion of the function f ( x, y ) = log( x ² + y ² ) at x = y = 1 in the form

f(x, y) = f (1, 1) + f _x (1, 1)(x − 1) + f _y (1, 1)(y − 1) + R 2 (x, y) (Express the error term R ₂ (x, y) in a suitable form).

(2) Draw the shape of the surface z = log(x ² + y ² ).

(3) Let x = y = 1. Determine one effective digit of the change of z = log( x ² + y ² ) if x increases by 0 . 003 and y decreases by 0 . 002. Also, explain the estimate of the error term on which your calculation is based.