LA1

Question 1

Vector

Answer 1

A vector is an element of a vector space, often represented as an ordered list of
numbers that indicate a direction and magnitude in space. Example:

a =[3]
[2]

Question 2

Matrix

Answer 2

A matrix is a rectangular array of numbers arranged in rows and columns.
Example:

A =[1 2]
[3 4]

Question 3

Subspace

Answer 3

A subspace is a set of vectors that is closed under vector addition and scalar
multiplication. Odd integers are not a subspace under the real numbers because
they do not satisfy closure under these operations: 3 + 3 = 6.

Question 4

Inverse of matrix

Answer 4

The inverse of a matrix A is another matrix A−1
such that:
A · A^−1 = I
where I is the identity matrix. The inverse is defined only for square matrices
with a non-zero determinant.

Question 5

Linear transformation

Answer 5

A linear transformation is a function between vector spaces that preserves vector
addition and scalar multiplication. It can be represented by a matrix A such
that T(x) = Ax.

Question 6

Formula for determinant of 2x2 matrix

Answer 6

For A =[a b]
[c d]:
det(A) = ad − bc
The determinant shows if a matrix is invertible and indicates scaling properties
of the transformation.

Question 7

What does it mean to span a space and what is a basis?

Answer 7

The span of a set of vectors is the set of all possible linear combinations of those
vectors. A basis is a set of linearly independent vectors that span a vector space.
A basis is orthonormal if all vectors are orthogonal and have unit length.

Question 8

What does it mean for two vectors to be linearly independent?

Answer 8

Two vectors are linearly independent if no scalar multiple of one can express
the other.

Question 9

What is the rank of a matrix?

Answer 9

The rank is the dimension of the vector space spanned by its rows or columns.