**MATH1025.03M**

**FW00**

**ANSWERS
TO TERM TEST 3A**

1. (4 marks) Find the standard matrices for the following linear transformations:

(a) T_{1}: R^{3 }->
R^{3}

w_{1} = 3x_{1}
- 4x_{2} + x_{3}

w_{2} = -x_{1}
+ 5x_{2} + 2x_{3} [T_{1}] =

w_{3} = 2x_{1}
+ 3x_{3}

(b) T_{2}: R^{3 }->
R^{2}

w_{1} = 4x_{1}
- 2x_{2} + 3x_{3} [T_{2}] =

w_{1} = 2x_{1}
- 5x_{2}

2. (4 marks) Determine whether the linear transformation in question 1(a) is one-to-one.

det[T_{1}] = 3x15 +
4x(-3 - 7) +1x(-10) = -5

Since this determinant is non-zero the transformation is one-to-one.

3. (5 marks)

(a) Find the composition T_{2}
o T_{1} of the two linear transformations in questions 1(a)
and (b).

[T_{2}oT_{1}]
= [T_{2}][T_{1}] =

(b) Is the composition in part (a) commutative? Justify your answer.

It is not commutative since
the matrix product [T_{1}][T_{2}] is not defined.

4. (2 marks) Write down the standard matrices for the following linear transformations:

(a) A reflection about the
line x = y in R^{2};

(b) A projection onto the
yz-plane in R^{3}.

5. (7 marks) Find the
standard matrix for the linear transformation T:R^{3} ->
R^{3} which reflects a vector in the xy-plane and then
projects it into the xz-plane by considering the effect of this
transformation on the standard basis **e**_{1},
**e**_{2}, **e**_{3}.

The linear transformations fave the following effect on the standard basis:

reflection in xy-plane projection into xz-plane

**e**_{1}
= (1, 0, 0) -> (1, 0, 0) -> (1, 0, 0) = T(**e**_{1})

**e**_{2}
= (0, 1, 0) -> (0, 1, 0) -> (0, 0, 0) = T(**e**_{2})

**e**_{3}
= (0, 0, 1) -> (0, 0, -1) -> (0, 0, -1) = T(**e**_{3})

Thus the standard matrix is
[T] = [T(**e**_{1}), T(**e**_{2}), T(**e**_{3})]
=

The following are the axioms
which define a **vector space** V where **u**, **v** and **w**
are vectors in V and k and m are scalars:

(1) If **u** and **v**
are in V then **u** + **v **is in V.

(2) **u** + **v **= **v**
+ **u**

(3)**
u** + (**v**
+ w)** **=** **(**u**
+ **v**) + **w**

(4) There is a zero vector **0**
in V such that ** u** + **0 **=**
0** + **u **for all **u** in V.

(5)
For each **u** in V there is a
negative -**u** such that
**u** + (-**u**) = (-**u**)
+ **u** = **0**

(6) For any **u** in V and
any scalar k, k**u** is in V.

(7) k(**u** + **v**) =
k**u** + k**v**

(8) (k + m)**u** = k**u**
+ m**u**

(9) k(m**u**) = (km)**u**

(10) 1**u** = **u**

6. (3 marks) Determine
whether the set of 2 by 2 matrices with **non-zero determinant**
is a vector space with the operation + denoting matrix multiplication
and the usual definition of scalar multiplication.

It is NOT a vector space since the following axioms do not hold:

(2) matrix multiplication is not commutative;

(6) fails when k = 0 since then the determinant of kA is zero

(7) fails since k(AB) is not equal to (kA)(kB)

(8) fails since (k+m)A is not equal to (kA)(mA)

It is sufficient to show one of these cases where an axiom fails in order to prove it is not a vector space.

7. (5 marks) Is the set of
vectors (x, 2x) a subspace of R^{2}? Justify your answer.

We must show that axioms 1 and 6 hold.

Let **u** = (x, 2x) and **v**
= (y, 2y).

Then **u** + **v** =
(x+y, 2x+2y) = (x+y, 2(x+y)) which is of the required form.

Also
k**u** = (kx, k(2x)) =
(kx, 2(kx)) which is of the desired form.

Thus the set of vectors of the form (x, 2x) is a subspace.

8. (5 marks) Are the vectors (2, -1, 5), (-1, 0, 3) and (4, 4, 2) linearly independent? Justify you answer.

The three vectors are linearly independent if

a(2, -1, 5) + b(-1, 0, 3) + (4, 4, 2) = (0, 0, 0)

has the only the trivial
solution a = b = c = 0. This is true if

has a non-zero determinant. The determinant has the value 2x(-12) + 1x(-2 - 20) + 4x(-3) = -58 so the vectors are linearly independent.

9. (2 marks) If V is any
vector space and S = {**v**_{1},
**v**_{2}, ...,
**v**_{n}} is a set of vectors in V, under what conditions
is S a **basis** for V?

It is a basis if:

i) the vectors in S are linearly independent;

ii) the vectors in S span the vector space V.

10. (7 marks) Find the
coordinates of 2 + 3x - x^{2} relative to the basis {1 + x, x
- x^{2}, 1 - 3x^{2}} for P_{2}.

Let a(1+x) + b(x-x^{2})
+ c(1-3x^{2}) = 2 + 3x - x^{2}. Thus

a + c = 2, a + b = 3, -b - 3c = -1

This system of linear equations has solution a = 2, b = 1, c = 0, i.e. the co-ordinates are (2, 1, 0).

11. (4 marks) Prove that
rank(A) = rank(A^{T}).

rank of A = dimension of column space of A = dimension of row space of A

But column space of A^{T }=
row space of A (or row space of A^{T} = column space of A).

Hence rank of A = rank of A^{T}.

12. (7 marks) Use the Gram-Schmidt method to transform the vectors {(1, 2, 3), (4, -1, 4)} into an orthonormal set.

Let **v**_{1}
= (1, 2, 3) and

**v**_{2}
= (4, -1, 4) - [(4, -1, 4).(1, 2, 3)/ **v**_{1}. **v**_{1}]
**v**_{1 }= (4, -1, 4) -[14/14] (1, 2, 3) = (3, -3, 1)

Normalize

13. (10 marks)

(a) Show that

is an orthonormal basis for
R^{3}.

Let the vectors be **v**_{1},
**v**_{2}. **v**_{3}. Then

Thus the set of vectors is
orthonormal and hence a basis for R^{3}.

(b) Find the coordinates of (2, -2, 5) with respect to the basis in part (a).

Let the vector be **v**.
Then the coordinates are

Thus the coordinates of **v**
with respect to this basis are