**MATH1025 W**

**PRACTICE
TEST WITH SOLUTIONS
**On Chapters 4, 5, and 6

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 operator 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 operator on the standard basis **e**_{1},
**e**_{2},
**e**_{3}.

The linear
operators have 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