MTAT.03.227 Machine Learning
Spring 2016 / Revision material
Scalar product and matrix algebra
Maximum score: HW6 ≤ 5p ⇒ 3p ∧ HW6 > 5p ⇒ 5p
Deadline: 7th of March 16:15 EET
1. Scalar product is defined ha, bi = |a|·|b|·cos φ where φ is the angle between
vectors a and b. What would happen if we would define co-scalar product
as hha, bii = |a| · |b| · sin φ?
(a) Show that co-scalar product is linear in two-dimensional vector space
R2 by generalising the argument given in the video lecture. Give an
analytical formula to evaluate co-scalar product (0.5p).
(b) Show that co-scalar product is not linear already in three-dimensional
vector space R3 . What is the main reason (0.5p)?
2. Use orthogonal decomposition for computing the volume of a four dimensional parallelepiped specified by Albrecht Dürer in 1514


16 3 2 3
 5 10 11 8 


 9 6 7 12 .
5 15 14 1
More precisely, interpret columns of the matrix as vectors in R4 and use
Gramm-Schmidt orthogonalisation procedure to compute (3 × 0.5p):
(a) area of a parallelogram defined by the first two vectors (16, 5, 9, 5)T
and (3, 10, 6, 15)T ;
(b) volume of a parallelepiped defined by the first three vectors (16, 5, 9, 5)T ,
(3, 10, 6, 15)T and (2, 11, 7, 14)T ;
(c) volume of a parallelepiped defined by all four vectors (16, 5, 9, 5)T ,
(3, 10, 6, 15)T , (2, 11, 7, 14)T and (3, 8, 12, 1)T .
How is this volume related to the matrix determinant? Do you get a
different answer if you interpret rows of the matrix as vectors?
3. Use orthogonal decomposition to find the point closest to the origin for
the plane defined by the row and column vectors of the following matrix


8 1 6
3 5 7 .
4 9 2
More precisely, interpret row and column vectors as points through which
the plane passes (0.5p + 0.5p). Are the distances different?
1
4. Verify that column vectors of the following matrix are ortogonal


1
1
1
1
1 −1
1 −1

 .
1
1 −1 −1
1 −1 −1
1
Treat column vectors as basis h1 , . . . , h4 and find corresponding coordinate representation xi = αi1 h1 + · · · + αi4 h4 for all column vectors of the
following matrix by crafted Josep Maria Subirachs


1 14 14 4
11 7 6 9 


 8 10 10 5  .
13 2 3 15
5. A scalar product is only one way to define orthogonality and vector lengths.
It is also possible use bilinear mapping hx, yiQ = xT Qy where Q is a
square matrix with right dimensions as a basis
pof geometry. In this case
the length of the vector is defined as kxkQ = hx, yiQ .
(a) The matrix Q can be arbitrary as long as it assigns non-negative
distances kxkQ for all vectors. Show that matrices
1
1
1
−4
and
1
−2
−2
1
are not suitable as there are vectors x for which kxk < 0 (0.5p).
Hint: You can still try to use orthogonalisation procedure to find examples of vectors with imaginary lengths. If the procedure completes
without errors, then the matrix Q is suitable.
(b) Find basis vectors that are orthogonal and compute unit Q-circle
{x : kxkQ = 1} for the matrix
2 −1
.
−1
2
Is it a coincidence that the Q-unit circle is an ellipse (0.5p).
2