**MATH1025.03M**

**FW00**

**ANSWERS
TO TERM TEST 3b**

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

(a) T_{1}: R2^{ }->
R^{3}

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

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

w_{3} = 2x_{1}
- 7x_{2}

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

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

w_{2}
= - 4x_{2} + 6x_{3}

w_{3}
= 3x_{1} + 5x_{2} + 7x_{3}

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

det[T_{2}] = 1(-28-30)
- 2(-8) + 3(12) = 14

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) Identify the linear transformations represented by the following standard matrices:

(a) Reflection about the x-axis

(b) Projection into the xz-plane

5. (7 marks) Find the
standard matrix for the linear transformation T:R^{3} ->
R^{3} which rotates a vector about the positive x-axis
clockwise through an angle pi/4 and
then reflects it about 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:

rotation about x-axis reflection about xz-plane

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

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

**e**_{3}
= (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 **=** 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) Consider the set of all vectors of the form (x, y) with the operation + denoted by

(x, y) + (u, v) = (xu, yv) and the usual definition of scalar multiplication.

(a) What is the zero vector defined in axiom 4) for this definition of 'addition'?

Since **u** + **0 **=**
**(x, y) + (0_{1}, 0_{2})
= (x0_{1}, y0_{2}) = (x, y) = **u this means
that 0 _{1} = 1 and 0_{2} = 1 i.e. the zero vector is
(1, 1).**

(b)
Whaaat is the negative vector defined in axiom 5) for this definition
of addition? Does it exist for all vectors (x, y)?** **

Since
**u **+ (-**u**)
= **0** or (x, y) + (-x,
-y) = (x(-x), y(-y)) = (1, 1) then -x = 1/x and -y = 1/y, i.e. -**u**
= (1/x, 1/y). This vector does not exist if x and/or y is zero.

7. (5 marks) Is the set of 2 by 2 matrices of the form

a subspace of M_{22}?
Justify your answer.

We must show that axioms 1 and 6 hold.

Let

A = and B =

Then

A + B = which is of the required form.

kA = which is of the required form.

Hence the set is a subspace.

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

The three vectors are linearly independent if

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

has the only the trivial solution a = b = c = 0.

This is true ifhas a non-zero determinant. The determinant has the value 3(2) - 2(6-4) + 1(1) = 3 so the vectors are linearly independent.

9. (2 marks) Are
the polynomials 1 + x^{2}, 2 - x a basis for P_{2}?
Justify your answer.

A basis for P_{2} must
have three basis vectors. Hence the given polynomials aaaer NOT a
basis.

10. (7 marks) Find the
coordinates of (2, 3, -5) relative to the basis (1, 0, -1), (2, -1,
0) and (0, 1, 2) for R^{3}.

We must find the coefficients a, b and c such that (2, 3, -5) = a(1, 0, -1) + b(2, -1, 0) + c(0, 1, 2)

or a + 2b = 2, -b + c = 3, -a + 2c = -5. This has the solution a = 13/2, b = -9/4 and c = 3/4 (the method of solution is not important) so that the co-ordinates are (13/2, -9/4, 3/4).

11. (4 marks) Prove that an nxn invertible matrix has rank n.

If A is invertible then A**x**
= **0** has only the
trivial solution which means the nullity of A is 0. But rank +
nullity = n, the number of rows of A, so that the rank = n.

12. (7 marks)

(a) For what value of s are the vectors (1, s, 3) and (2, -2, s) orthogonal?

For the vectors to be orthogonal their dot product must be zero, i.e. 2 - 2s + 3s = 0 or s = -2.

(b) Find a vector which is orthogonal to the two vectors in part (a).

Let the vector be (x, y, z). Then x + sy + 3z = 0 and 2x - 2y + sz = 0 with s = -2. Solving this system yields z = t, y = 4t, x = 5t so the vectors is t(5, 4 1).

13. (10 marks) Use the
Gram-Schmidt method to transform the vectors (1, 3) and (2, 2)} into
an orthonormal basis for R^{2}.

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

Then
take **v**_{1} =
(1, 3) and **v**_{2}
= **u**_{2} - (**u**_{2}
. **v**_{1})/(v_{2}
. **v**_{1}) **v**_{1}
= (2, 2) - (8/10)(1, 3) = 6/5, -2/5)

Normalize

(b) Find the coordinates of (2, -2) 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