Find out if the vectors (0, 1, 0, 1), (0, 0, 2, 0), (1, 0, 1, 0) and (0, 1, 0, 2) form a basis for R⁴. If this is so, find out the coordinates of (1, 2, 5, 5) relative to this basis.

An example
Answer

S₁ = {(0, 1, 1), (0, 0, 1), (1, 1, 1)} and S₂ {(1, 2, 0), (1, 0, 0), (1, 2, 3)} are both bases for R³. The coordinates of a vector relative to the latter basis are. -1, 2, 3. Compute the basis change matrix and find out the coordinates of relative to the former basis.

An example
Answer

In computer aided design (CAD) linear transformations such as stretching, rotations and reflections are used. Determine the matrix of the linear transformation (relative to the standard basis E = , , ), when an image is first stretched 3-fold in the direction of the j-axis , then stretched 2-fold in the direction of the k-axis and then rotated counter-clockwise (viewed from the positive k-axis to the origin) about the k-axis by an amount of /2 radians. What is the matrix if we also reflect the image in such a way that the inversion plane is the xz- plane ( = ik-plane) and then rotate clockwise (viewed from the positive j--axis to the origin) about the j-axis through an angle of 3/2?

An example
Answer

Again, consider image processing in CAD. May the order of two streching or two rotations be interchanged? How about a rotation and a stretching?
Reason your answer.

Answer

Compute the matrix of the linear mapping F: R² -> R³,

relative to the bases {( 1, 1), (0, -1)} and {( 1, 0, 1), (0, 1, 0), (0, 1, 1)}.

An example
Answer

Compute the matrix of the linear mapping F: R³ -> R²,

relative to the bases {( 1, 1, 1), (0, 1, 1), (0, 0, 1)} and {( 1, 1), (1, 0)}.

An example
Answer

Exercise 29.

Find the rank, the nullity, the kernel and a basis for the kernel of the following matrices:
a)

	1	1	0		,
	0	1	-1
	1	0	1

b)

	-1	2	1	-1
	1	3	2	-3
	2	1	-1	2
	4	-2	-2	6

c)

	1	1	1	1		.
	1	2	1	2
	2	3	2	3

An example
An example
Answer

---

Exercise 30.

Consider the matrix

A =		1	2	-1	2
		2	5	-2	3
		-1	-3	1	-1

Find
a) the rank and the nullity,
b) the kernel and a basis for the kernel,
c) a basis for the image.

Answer

---

Exercise 31.

Use determinant to find the equation
a) of a plane passing through the points (2, 3, 1), (2,-1, -1) and (1, 2, 1),
b) of a circle passing through the points (2, 6), (2, 0) and (5, 3).

An example
An example
Answer

---

Exercises 32-35
Contents
Index
All exercises

---