MATH1071-WE01 Linear Algebra I 2021
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MATH1071-WE01
Linear Algebra I
2021
Q1 1.1 Find three linear equations in three variables, i.e., equations of the form ax +by +cz = d with a, b, c, d ✷ ❘ , such that any two of them have an infinite number of common solutions, but all three have no solution in common. Briefly justify your answer.
1.2 Let L1 and L2 be two non-intersecting, non-parallel lines in ❘3 , neither of which pass through the origin. Consider the set
X = ♥x ✷ ❘3 ❥x = u + v for some u ✷ L1 and v ✷ L2 ♦ .
Depending on L1 and L2 , which of the following may be true? For those you think can happen, give an example; for those you think cannot happen, prove your assertion. Justify your answers carefully.
(i) X is the whole of ❘3 ;
(ii) X is a 2 dimensional vector subspace of ❘3 ;
(iii) X is a 2 dimensional affine subspace of ❘3 ;
(iv) X is a 1 dimensional vector subspace of ❘3 ;
(v) X is a 1 dimensional affine subspace of ❘3 ;
(vi) X is none of the above.
Q2 2.1 For what values of s, t ✷ ❘ is the matrix
A = ❇ s 0
❅ 1 t
2✶
1❆
invertible? Justify your answer, including showing any working and stating any results you use from the lecture notes.
2.2 Now let t = 0. For those values of s for which you think the matrix is invertible, use the Gauss-Jordan algorithm to find the inverse of A as a function of s. Show your working fully.
Q3 The set A of vectors given below span a subspace V of ❘4 . Find a basis for the orthogonal complement of V (using the standard inner product on ❘4 ), and find a subset of A which is a basis for V, justifying that it is a basis.
✽❃❃ ✵ 1✶ ✵ 2 ✶ ✵ 2 ✶ ✵3✶ ✾❃❃
A = ❃❁❃ 0(1) , 31 , 63 , 1(1) ❃❂❃ .
❃❃✿ ❅2❆ ❅ 1 ❆ ❅ 3 ❆ ❅ 1❆ ❃❃❀
Q4 4.1 Let ❘[x]n be the vector space of polynomials in x with real coefficients and
degree at most n. Define the function A : ❘[x]n ✦ ❘[x]n by
A(f(x)) = f(x) f(x + 1) .
Prove that A is a linear map.
4.2 Find a basis for each of the kernel and the image of A, justifying your answer. Now describe the image of Am (the composite of A with itself, m times) for each positive integer m, explaining your reasoning.
✵ 1✶ ✵ 0✶
Q5 5.1 Let ❢e1 , e2 , e3 , e4 ❣ be the standard basis for ❘4 , that is, e1 = 0(0) , e2 = 0(1) ,
❅ 0❆ ❅ 0❆
✵ 1✶ ✵ 1✶
etc. Let x1 = ❇❇❇❅❈❈❈❆ and x2 = ❇❇❇❅❈❈❈❆. Use the process applied in the proof of
the Steinitz Exchange Theorem (Theorem 6.2.9 of the Michaelmas lecture notes) to find a basis for ❘4 containing x1 , x2 and some of the ei . Do this three times to obtain three distinct bases. Show your working.
5.2 Suppose the finite dimensional real vector space V is the direct sum U1 ✟U2 of subspaces U1 and U2 . Suppose given linear maps f1 : U1 ✦ W and f2 : U2 ✦ W. Form the linear map F : V ✦ W by
F(u1 + u2 ) = f1 (u1 ) + f2 (u2 ) for u1 ✷ U1 , u2 ✷ U2 .
If fi has rank ri , express the rank of F in terms of r1 , r2 and any other number or numbers you consider relevant, and state what the full range of possible values this rank may take. Briefly explain your reasoning, including stating any results you use from lectures.
Q6 6.1 Find all the values of r ✷ ❘ for which the matrix
Ar = ✥ 0(0) r(1)✦
is diagonalizable.
6.2 For the same matrices Ar as in the previous question, find all the values of r ✷ ❘ such that Ar is orthogonally diagonalizable.
6.3 Give an example for every integer n ❃ 2 (justifying your answer) of a non- diagonalizable n ✂ n matrix Bn such that
n
Bn(2) ❨ (Bn kI) = Bn(2)(Bn 3I)(Bn 4I) ✁✁✁(Bn nI) = 0 .
=3
Q7 Consider the subspace U of ❘3 (with its standard inner product) given by vectors (x, y, z) satisfying
2x y + z = 0 .
Q8 Consider the vector space Symn of real symmetric n ✂ n matrices.
8.1 Show that
(u, v) =❳❳ uij vji
i=1 j=1
for u, v ✷ Symn defines an inner product on Symn .
8.2 Consider the case of real symmetric 2 ✂ 2 matrices Sym2 (with the inner product as above), and define D2 to be the subspace of diagonal matrices. Find the matrix in D2 nearest to
w = ✥ 1(0) 0(1)✦ .
Q9 Consider the vector space ❘[x] with inner product
(f, g) = ❩11 (1 x2 )f(x)g(x)dx
and the family of differential operators ▲k : ❘[x] ✦ ❘[x]
d2 d
dx2 dx
with k ✷ ❘ .
9.1 Show that ▲k is a symmetric operator if and only if (▲k (x2 ), 1) = (x2 , ▲k (1)). Find the set S = ❢k such that ▲k is symmetric❣ .
9.2 Restricting yourself to the space of quadratic polynomials ❘[x]2 , compute the matrix representation, eigenvalues and eigenvectors of ▲✛ for all ✛ ✷ S.
Q10 Define the set of matrices
✽❃✵0 0 0✶ ✾❃
❃✿❅0 0 1❆ ❃❀
Determine whether G together with the following group operations is a group.
(i) Matrix multiplication.
(ii) Matrix addition.
(iii) The product H(A, B) defined for A = (aij ), B = (bij ) by H(A, B)ij = aij bij .
2022-04-08