Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Digital systems, linear algebra and introduction to formal proof / 4 / e

Table of contents
- Solution e

Solution

Both

L

and

R

have

m

linearly independent columns. The Gram matrix

R^{⊤} R

is invertible because

y^{⊤} R^{⊤} R y = ∥ R y ∥^{2} > 0 (y \neq = 0) .

Consequently

R^{⊤}

is a surjective linear map: for any

y \in R^{m}

x = R (R^{⊤} R)^{- 1} y

satisfies

R^{⊤} x = y

. Hence

im B = L (im R^{⊤}) = im L .

The latter image of a linear map has dimension

m

, giving $rank B = m$ . If

m = 0

,

B = 0

and the conclusion is immediate. This argument explains why the product retains its full possible matrix rank; a general product of two matrices need not do so.

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