COMP0142 Machine Learning For Domain Specialists
Topic 2.1: Linear Algebra Questions With Complete
Solutions
(Differential) Calculus Correct Answers Minimising cost
functions (a scalar function of several variables that typically
measures how poorly our model fits the data) to study their
continuous change
**Affine subspaces (Linear manifold) Correct Answers ** The
subset L of V such that L = x0 + U (where U is a subspace of V,
called the **direction space**, and x0 is a vector in V, called
the **support point**), i.e. the set {**v**∈*V* |
∃**u**∈*U* : **v**=**x**0 +**u**}. Note that when x0 is
not in U, it will exclude the null vector 0 so it is **not a vector
space.** Affine subspaces essentially translate the origin, or
look at a vector space from another perspective/viewpoint. All
operations in V have a corresponding one in A since the map
v→a+v is a bijection
**argmin Correct Answers function to find the smallest
possible value for a function**
**Linear map (not all functions are linear maps, e.g. squaring
function is not) Correct Answers ** a function which preserves
the operations of vector addition and scalar multiplication
(additivity and homogeneity of degree)
**Nullity Correct Answers dim(null(A)) the dimension of the
null space of A**
, **Parametric subspace Correct Answers ** You can define the
affine subspace, L, by parameters **x** = **x**0 + λ1**x**1
+ . . . + λ*k* **x***k* where {x1, ... xk} is a basis of U. We
can use this to describe lines, planes and (n-1)-dimensional
hyperplanes in vector space Rn.
**Rank (Note Correct Answers Rank of A = Rank of A
transpose): dimension(range(A)) =
dim(columspace)=dim(rowspace)= no. of linearly independent
vectors amongst the columns (row) of A ≤ min(n,m)**
**Rank-Nullity Theorem Correct Answers rank(A) + nullity(A)
= m (number of columns)**
Additivity Correct Answers f(x+y) = f(x) + f(y)
Affine Correct Answers avoid the origin
Affine maps Correct Answers linear maps plus translation (i.e.
may move the origin). Different perspectives are related by
translation but may have really different answers to similar
questions. [https://youtu.be/c-iA43vPL9M](https://youtu.be/c-
iA43vPL9M)
Augmented matrix Correct Answers a matrix obtained by
appending the columns of two given matrices, usually for the
purpose of performing the same elementary row operations on
each of the given matrices
Basis for V Correct Answers a set of linearly independent
vectors which span the whole of V
Topic 2.1: Linear Algebra Questions With Complete
Solutions
(Differential) Calculus Correct Answers Minimising cost
functions (a scalar function of several variables that typically
measures how poorly our model fits the data) to study their
continuous change
**Affine subspaces (Linear manifold) Correct Answers ** The
subset L of V such that L = x0 + U (where U is a subspace of V,
called the **direction space**, and x0 is a vector in V, called
the **support point**), i.e. the set {**v**∈*V* |
∃**u**∈*U* : **v**=**x**0 +**u**}. Note that when x0 is
not in U, it will exclude the null vector 0 so it is **not a vector
space.** Affine subspaces essentially translate the origin, or
look at a vector space from another perspective/viewpoint. All
operations in V have a corresponding one in A since the map
v→a+v is a bijection
**argmin Correct Answers function to find the smallest
possible value for a function**
**Linear map (not all functions are linear maps, e.g. squaring
function is not) Correct Answers ** a function which preserves
the operations of vector addition and scalar multiplication
(additivity and homogeneity of degree)
**Nullity Correct Answers dim(null(A)) the dimension of the
null space of A**
, **Parametric subspace Correct Answers ** You can define the
affine subspace, L, by parameters **x** = **x**0 + λ1**x**1
+ . . . + λ*k* **x***k* where {x1, ... xk} is a basis of U. We
can use this to describe lines, planes and (n-1)-dimensional
hyperplanes in vector space Rn.
**Rank (Note Correct Answers Rank of A = Rank of A
transpose): dimension(range(A)) =
dim(columspace)=dim(rowspace)= no. of linearly independent
vectors amongst the columns (row) of A ≤ min(n,m)**
**Rank-Nullity Theorem Correct Answers rank(A) + nullity(A)
= m (number of columns)**
Additivity Correct Answers f(x+y) = f(x) + f(y)
Affine Correct Answers avoid the origin
Affine maps Correct Answers linear maps plus translation (i.e.
may move the origin). Different perspectives are related by
translation but may have really different answers to similar
questions. [https://youtu.be/c-iA43vPL9M](https://youtu.be/c-
iA43vPL9M)
Augmented matrix Correct Answers a matrix obtained by
appending the columns of two given matrices, usually for the
purpose of performing the same elementary row operations on
each of the given matrices
Basis for V Correct Answers a set of linearly independent
vectors which span the whole of V
