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The rational number22has been used as an approximation to the number r since the time of Archimedes. Show that

\int_{0}^{1} \frac{x^{4}(1-x)^{4}}{1+x^{2}} d x=\frac{22}{7}-\pi

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Vectors

Suppose that

\mathbf{A}=\left(\begin{array}{cccc}
1 & x & y & 1 \\
1 & x & x & x \\
x & 1 & x y & y \\
x & x & x y & 1
\end{array}\right) \in \mathscr{M}_{4,4}(\mathbb{R})

Show that A is not invertible if and only if either x = y, x =1 or xy = 1.

Let V and W be 3-dimensional vector spaces over R, Bị = {V1, V2, V3} be an ordered basis for V and B2 = {w1,w2, w3} be an ordered basis for W. Define a linear transformation T :V → W by

T\left(\mathbf{v}_{1}\right)=\mathbf{w}_{1}+\mathbf{w}_{2}

T\left(\mathbf{v}_{2}\right)=\mathbf{w}_{1}-\mathbf{w}_{2}+\mathbf{w}_{3}

T\left(\mathbf{v}_{3}\right)=\mathbf{w}_{3}

(i) Write down the matrix A representing T with respect to the ordered bases B1 and

(ii) Calculate the dimension of the kernel of T and of the image of T.

(iii) Show that B3 = {v1,V1 + V2, V1 + V2 + V3 } is a basis for V and B4 = {w1 -

W2, W2 +W3, W1 – W3} is a basis for W.

) Find the matrix which represents T with respect to the ordered bases B3 and B4.

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Vectors

Prove that

\delta\left[a\left(x-x_{1}\right)\right]=\frac{1}{a} \delta\left(x-x_{1}\right) .

\text { Note. If } \delta\left[a\left(x-x_{1}\right)\right] \text { is considered even, relative to } x_{1} \text {, the relation holds for negative } a

and 1/a may be replaced by 1/la|.

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Vectors

Problem 6. (Exercise 6 in the lecture on6.23) Finish the proof of either of the following statements:

(a) Let S =then S contains a subset S', such that S' is a basis ofV.{v1, v2, ..., Um} be a spanning set of a finite dimensional F-vector space V,

(b) Let S = {v1, V2, . .. , Um} be a linearly independent set of a finite dimensional F-vector space V, then S is a subset of some S' C V, such that S' is a basis of V.

Using the statement that you proved, show that if W is a subspace of V, then dim W <dim V.

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Vectors

Problem 5. (Exercise 5 in the lecture on 6.23) Let V be an F-vector space, {v1,..., Vm} be a list of vectors in V. Show that the two definitions of linear independence are equivalent:

(a) For any i = 1,..., m, v; is not a linear combination of the rest of the vectors in the list.

\text { (b) Let } a_{1}, \ldots, a_{m} \in \mathbb{F} \text {, then } a_{1} v_{1}+a_{2} v_{2}+\ldots+a_{m} v_{m}=0 \text { if and only if } a_{1}=\ldots=a_{m}=0 \text {. }

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Vectors

Problem 3. (Exercise 4 in the lecture on 6.23) Let F be a field and Fm[r] be the set ofpolynomials with coefficients in F whose degree is less than m. Show that Fm[x], equippedwith the usual addition and scalar multiplication, is a vector space over F. In the lecture on6.23, we will show that F[r], the set of polynomials of arbitrary degrees with coefficients in F,is a vector space. So you only need to show that Fm[r] is a subspace of F[r] for any positiveinteger m.

Problem 3. (Exercise 4 in the lecture on 6.23) Let F be a field and Fm[z] be the set ofpolynomials with coefficients in F whose degree is less than m. Show that Fm[r], equippedwith the usual addition and scalar multiplication, is a vector space over F. In the lecture on6.23, we will show that F[r], the set of polynomials of arbitrary degrees with coefficients in F,is a vector space. So you only need to show that Fm[r] is a subspace of F[r] for any positiveinteger m.

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Vectors

Problem 1. (Exercise 1 in the lecture onfield:6.21) Using the definition of fields, show that in a

(a) The zero scalar is unique,

(b) The additive inverse of any scalar is unique,

(c) The unit scalar is unique,

(d) The multiplicative inverse of any nonzero scalar is unique.

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Vectors

k + 0 is an eigenvalue with corresponding eigenvector (1, 2, 3) of

A=\left[\begin{array}{ccc}
-6 & -2 & 3 \\
c & a & 4 \\
-3 & -6 & b
\end{array}\right]

Find a, b, k.

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Vectors

\text { For the matrix } A=\left[\begin{array}{rrr}
4 & 8 & 0 \\
-1 & -2 & 0 \\
0 & 0 & 2
\end{array}\right] \text { find all the eigenvalues and for each }

eigenvalue the general solution to the eigenvector problem.

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Vectors

\text { a) The matrix } Y=\left[\begin{array}{lll}
0 & 1 & 2 \\
0 & 0 & 4 \\
0 & 0 & 0
\end{array}\right] \text { has } 0 \text { as its only eigenvalue. }

Find the general solution to the problem of finding the corresponding eigenvectors.

\text { The matrix } Z=\left[\begin{array}{lll}
1 & 2 & 3 \\
0 & 1 & 2 \\
0 & 0 & 1
\end{array}\right] \text { has } 1 \text { as its only eigenvalue }

Find the general solution to the problem of finding the corresponding eigenvectors.

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Vectors

\text { For the matrix } X=\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right] \text { find the eigenvalues and an eigenvector }

corresponding to each.

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