MATH2501 Linear Algebra Term 2 2020
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Term 2 2020
MATH2501
Linear Algebra
1. a) [3 marks] Let Ax = b be a system of linear equations, where A e Mmn (R) and b e Rm . Let ╱ A b 、be the augmented matrix of the system and let ╱ U y 、be the corresponding row echelon form (REF).
Using the terms leading column and non-leading column and using the REF com- ponents U and y, explain when the system Ax = b has
i) no solutions;
ii) unique solution;
iii) multiple solutions.
b) [5 marks] Let a e R. Consider the following system of linear equations
_2x1 +4x2 +2x3 = 2
ax1 +2x2 _x3 = 1
i) Find the values of the parameter a such that the system of equations above has multiple solutions.
ii) For every value of a in the preceding part, find all of the solutions in vector form. Hint: You may use the maple output show in Fig.1 at the end of this exam paper. Make sure you reference individual output lines in your solution.
c) [4 marks] Let V = (V, +, ., R) be a vector space, let B = ,v1 , v2 、be a basis and let a e V .
i) Explain what it means that the couple (a1 , a2 ) is the co-ordinate vector of a with respect to the basis B . Hint : Possible answer may start as follows:
A couple (a1 , a2 ) e R2 is the coordinate vector of a e V if and only . . .
ii) Explain what the symbol [a]B represents. Hint : Possible answer may start as follows:
If the couple (a1 , a2 ) e R2 is the coordinate vector of a e V then [a]B = . . .
iii) Prove that if a = 0, then [a]B = (0, 0). Name the quality of the basis B which is essential for the proof.
d) [2 marks] Let V = (V, +, ., R) and (W, +, ., R) be vector spaces. Define what it means that the map T : V → W be linear. Hint : Possible answer may start as follows:
A map T : V → W is called linear if and only if . . .
e) [6 marks] Let V and W be vector spaces. Let T : V → W be a linear map, let B = ,v1 , v2 、be a basis in V and let C = ,w1 , w2 、be a basis in W .
i) Define what it means that the matrix A e M2 ,2 (R) is the matrix of T with respect to to basis B in V and basis C in W . Hint : Possible answer may start as follows: The matrix A e M2 ,2 (R) is called the matrix of T with respect to to basis B in V and basis C in W if and only if
[T (x)]... = . . .
ii) Let
0(1)( e M2 ,2 (R)
be the matrix of T with respect to basis B in V and the basis C in W .
A) Find T (v1 ) and T (v2 ) in terms of elements of C .
B) Let D = ,u1 , u2 、be another basis in W and let
w1 = u1 + u2 and w2 = u1 _ u2 .
Find [T (v1 )]D and [T (v2 )]D .
C) Find the matrix of T with respect to the basis B in the domain and the basis D in the co-domain.
2. Below, the term scalar product is synonym to the term inner product. Let V be a vector space with scalar product V = ╱V,〈., .|、.
a) [3 marks] Explain what it means that the function〈., .| : V × V → R is positive- definite. Hint : Possible answer may start as follows:
The function〈., .| : V × V → R is called positive- definite if [. . . ]
b) [2 marks] Give an example a scalar product in R3 ,〈., .| : R3 × R3 → R, which is not the dot product. Prove that function in your example is positive-definite. You do not have to verify any other axiom of the scalar product.
c) [1 mark] Let x, y e V . Explain what it means that the vectors x and y are perpen- dicular, i.e., x 1 y. Hint : Possible answer may start as follows:
Two vectors x, y e V are perpendicular if and only if [. . . ]
d) [6 marks] Let x, y e V and let x 1 y. Prove the Pythagoras Theorem
|x + y|2 = |x|2 + |y|2 .
Hint : A possible proof may use the facts given below. Make sure you copy the facts into your work filling in the missing expressions. Make sure you annotate the steps in your proof where each fact i, ii, iii is used.
i) Scalar product is linear with respect to the first argument, i.e., ′a + b, c\ = . . ., for any a, b, c e V .
ii) Scalar product is linear with respect to the second argument, i.e., ′a, b + c\ = . . ., for any a, b, c e V .
iii) Scalar product is commutative, i.e., ′a, b\ = . . .
e) [1 mark] Write the statement the Cauchy-Schwartz Inequality. Hint : Possible answer may start as follows:
Let (V〈., .|) be a vector space with scalar product. For any vectors x, y e V , the Cauchy- Schwartz Inequality states that . . .
f) [2 marks] Let V = P5 (R) and let the scalar product be given by
1
〈p, q| = p(x)q(x) dx.
- 1
Copy the following statement into your work and replace every appearance of [ . . .] with correct expressions:
The Cauchy-Schwartz Inequality in P5 (R) states that, for every p, q e P5 (R), the fol- lowing inequality holds
│ -11 p(x) . . . │ < ← -11 ╱p(x)、2 . . . × [. . .].
g) [5 marks] Let S C V be subset.
i) Give definition of the orthogonal complement SL . Hint: Possible answer may start as follows:
Let (V〈., .|) be a vector space with scalar product and let S C V be a subset. The orthogonal complement is defined by [. . . ]
ii) Let x e V . Prove that, if x e S and x e SL , then x = 0. Make sure you carefully reference every scalar product axiom you use in your proof.
3. a) [1 mark] Let Q e Mnn (R). Explain what it means that the matrix Q is orthogonal. Hint : Possible answer may start as follows:
A matrix Q e Mnn (R) is called orthogonal if and only if . . .
b) [4 marks] Let α e R and let
Qα = ( e M2 ,2 (R)
i) Prove that Qα is orthogonal.
ii) Let x, y e R2 and let x 1 y. Prove that Qα x 1 Qα y. Make sure that you carefully state on the margins of your work, every trigonometric identity and/or any other result you use in your proof.
c) [7 marks] Let A, D, P e M2 ,2 (R) such that
D = diag ,_2, 1、, P = ( and A = PDP- 1 .
i) Explain why the following equation describes a hyperbola. xT Ax = 1, x = ! (
ii) Find the vectors a1 and a2 for the hyperbola above as shown on Fig. 3.
iii) Find the vectors u1 and u2 for the hyperbola above as shown on Fig. 3 in the direction of asymptotes of the hyperbola. Note: You do not have to normalise the vectors u1 and u2 .
d) [1 mark] Define what it means that a matrix A e Mnn (R) is invertible. Hint : Possible answer may start as follows:
A matrix A e Mnn (R) is invertible if and only if there is a . . .
e) [7 marks]
i) Explain what it means that a polynomial m(x) e Pn (R) is the minimal polynomial for the matrix A e Mnn (R). Hint : Possible answer may start as follows: A polynomial m(x) e Pn (R) is called the minimal polynomial of a matrix A e Mnn (R) if and only if . . .
ii) State the Minimal Polynomial Theorem. Hint : Possible answer may start as follows: If m(x) e Pn (R) is the minimal polynomial of a matrix A e Mnn (R) and if pA (z) . . . , then . . .
iii) Let m(z) = z2 _ 1 be the minimal polynomial of the matrix A e M5 ,5 (R). Prove that A is invertible.
4. a) [3 marks] Let A and B be two n × n matrices.
i) Define what it means that the matrices A and B are similar.
ii) Prove that if A and B are similar, then A _ λI and B _ λI are similar for every λ e C.
iii) Prove that if A _ λI and B _ λI are similar for every λ e C, then A and B are similar.
b) [5 marks] Let a, b, c e R and let A e M3 ,3 (R) be given by
!a _ b _ c 2a 2a !
A = . 2b b _ c _ a 2b .
{ ì
i) Prove that
! 1
B = . 2b
{
ii) Prove that
_a 00_b _ c │
iii) Find det B and det A.
c) [5 marks] An engineer is trying to fit a simple wave form to a signal. She has measured the following responses xi at the given times ti
|
1 |
2 |
3 |
4 |
ti |
0 |
π |
π |
π |
xi |
1 |
^3 |
1 2 |
1 _ |
The engineer tries to fit a curve of the form x = a cos(t) + b sin(t) to this data in the least squares sense.
i) Find the matrix A e M4 ,2 (R) and the vector b e R4 such that
i1 │xi _ a cos ti _ b sin ti │2 =冂(冂)Ax _ b 冂(冂)2 , where x = ! (b(a)
ii) If A e M4 ,2 (R) and y e R4 . Write the formula for the vector x e R2 which minimises the expression
冂(冂)Ax _ b 冂(冂)2 .
iii) Find the values a and b such that the square error of approximation
4
│xi _ a cos ti _ b sin ti │2
j=1
is minimal. Hint: You may use the maple output shown in Fig. 2 at the end of this exam paper.
d) [7 marks] Let nj e N, j = 1, 2, 3, 4 and let
n1 < n2 < n3 < n4 .
Let F e M14 , 14 (R) be such that
F ~ Jn1 (5) o Jn2 (5) o Jn3 (5) o Jn4 (5).
Let
dk = nullity ╱F _ 5I14、k , k e N.
i) Find n1 + n2 + n3 + n4 .
ii) Find the value d1 .
iii) What is the relation between d1 and the geometric multiplicity of the eigenvalue 5?
iv) Let d2 = 8 and d3 = 10. Find n1 and n2 .
v) Find all possible values of n3 and n4 under the assumption of the subpart (iv).
Figure 1: Maple Output for Q1b
Figure 2: Maple Output for Q4c
Figure 3: Hyperbola
2022-08-10