Vector spaces and matrices question

In summary, the conversation discusses the verification of the real (2x2)-matrices as a vector space with standard addition and multiplication by real scalars. It also mentions finding a basis for this vector space and determining its dimension.
  • #1
sheelbe999
13
0
Massively stuck with this one, have done some reading and am having difficulty connecting matrices to vector spaces

(a) Verify that the space of the real (2 x 2)-matrices, endowed with the standard addition
and multiplication by real scalars, forms a vector space
(b) Specify a basis for this vector space
(c) What is its dimension?

any help would be greatly appreciated.
 
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  • #2
For (a), you must show that the properties of a vector space are satisfied. See: http://en.wikipedia.org/wiki/Vector_space#Definition

For (b), find a set that spans this vector space (ie any vector in the space can be written as a linear combination of vectors in your spanning set). After you have done this, try to find the minimal number of vectors that can accomplish this. See: spanning set, linear independence

For (c), The dimension of the vector space is the number of vectors in the basis.
 

Related to Vector spaces and matrices question

What is a vector space?

A vector space is a mathematical concept that represents a set of objects, called vectors, which can be added and multiplied together to produce new vectors. It is a fundamental concept in linear algebra and has many applications in physics, engineering, and computer science.

How do you determine if a set of vectors form a vector space?

To determine if a set of vectors form a vector space, you need to check if they satisfy the 10 axioms, or properties, of a vector space. These include properties such as closure under addition and scalar multiplication, existence of a zero vector and additive inverse, and associativity and distributivity. If all 10 axioms are satisfied, then the set of vectors form a vector space.

What is a matrix and how is it related to vector spaces?

A matrix is a rectangular array of numbers or symbols arranged in rows and columns. It is closely related to vector spaces because a vector space can be represented by a set of vectors, and these vectors can be organized into a matrix. Matrices can also be used to represent linear transformations between vector spaces.

How do you perform operations on matrices, such as addition and multiplication?

Addition and subtraction of matrices are performed by adding or subtracting corresponding elements in the matrices. Multiplication of matrices is more complex and involves multiplying rows and columns of the matrices together. It is important to note that the dimensions of the matrices must be compatible for these operations to be performed.

What are some applications of vector spaces and matrices in real life?

Vector spaces and matrices have many practical applications in fields such as engineering, physics, computer graphics, and economics. They are used to model physical systems, solve systems of equations, compress and encrypt data, and perform transformations in 3D graphics. They are also used in machine learning and data analysis to represent and process large datasets.

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