A1=[ 1 0 1 0 ], A2=[ 1 1 0 0 ], A3=[ 1 0 0

A1=[ 1 0 1 0 ], A2=[ 1 1 0...

Key Concepts

Vector Space of Matrices

Matrix spaces, such as the set of all 2×2 matrices, form vector spaces where matrices play the role of vectors. This concept involves operations like matrix addition and scalar multiplication, and the space satisfies the vector space axioms. It is foundational in understanding how matrices can be manipulated and combined in linear algebra, and how they can represent transformations or systems of equations.

Basis of a Vector Space

A basis is a set of linearly independent elements in a vector space that spans the entire space. In the context of matrices, a basis allows us to express any matrix (from that space) uniquely as a linear combination of the basis matrices. This concept is critical for determining the dimensions of the vector space and for performing transformations between different coordinate representations.

Linear Combination

A linear combination involves expressing an element of a vector space—in this case, a matrix—as the sum of scalar multiples of basis elements. This operation is at the heart of many linear algebra techniques, such as solving linear systems, diagonalization, and change of basis, and it provides a systematic way to rebuild any element in the space from a given set of building blocks.

Linear Independence

Linear independence is a property of a set of vectors (or matrices) where none of the elements can be expressed as a linear combination of the others. This concept ensures that the basis provides a unique representation for every element in the vector space and is essential when choosing or verifying a basis in any vector space.

Transcript

00:01 Hello there.

00:02 For the following exercise, we need to consider the vector space defined by the matrices, the square matrices of dimension 2.

00:11 In this case, we have this set s, composed by these four matrices, and we have also this matrix a that is defined by 6, 2, and 3.

00:25 Now the first thing that we need to do is to show that s is a basis for this space.

00:31 And once that we have to find that s is a basis, we need to find the coordinate vector of the matrix a relative to this basis.

00:42 Okay, so let me start by showing that s is a basis.

00:49 We need to remember that a basis should satisfy two axioms.

00:56 Okay one is that the set s should be linearly independent and the other is that the span of s it's equal to the vector space if s or this vector in this case or any vector satisfy these two conditions then we are we can call that s is a basis okay so what's the meaning of linearly independence and as a pun, well we can represent both as a system of equations.

01:37 Okay, so let's first look what is linearly independence.

01:48 So linearly independence, if you remember, means that a linear combination of the basis elements it's equal to the zero vector in the vector space, in this case is the zero matrix, if and only if all these coefficients c1, c2, c3 and c4 are equals to 0.

02:22 Otherwise that means that one of the elements in the set is a linear combination of the other three, or in this case.

02:38 Okay, so we have this linear combination that is equal to the zero vector.

02:45 And the condition for the span means that any vector or any matrix in this case of the form b1, b2, b3, b4 element of the vector space can be written as a linear combination of the basis elements.

03:10 Or in expression that means that this linear combination should be equal to that element in the vector space.

03:25 So we need to find a unique solution for the system.

03:37 A unique c1, c2, c3 and c4.

03:46 Okay, so you can observe the weekend represents linear independence and spanish.

03:52 Of the space by these two equations.

03:57 But these two equations can be represented as a linear system.

04:04 So look that i'm going to focus on these linearly independence, and then from that it will be clear what happened with the span.

04:16 Okay, so let's consider just the linearly independence condition.

04:23 For matrices you need to focus on each element in the matrix.

04:29 So let's say the entry m111 that corresponds to this position in all the matrices, all of them should be the same.

04:39 Okay, we need to equate that position of the matrices, in other words.

04:45 So that means that here we have c1 plus c2 plus c3.

04:52 Equals to 0.

04:55 The same with the second position m1 for the position m12.

05:00 In that case, we need to consider this position, we need to equate what we have in all those positions for these matrices...

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