Find a basis for the space of 2×2 lower triangular matrices Basis =

Question

Find a basis for the space of 2×2 lower triangular matrices  Basis =

Thomas Conway · Accepted Answer

S = set of lower triangular matrices. To find basis of s Let A=[a0bc]ϵS A=a[1000]+b[0010]+c[0001] ∴ A can be written as the linear combination of vectors [1000],[0010]and[0001] and B=[1000],[0010],[0001] is linearly independent set ∴ B is basis for the space of 2×2 lower triangular matrices Finally answer: B=[1000],[0010],[0001]

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

star233 · Accepted Answer

Step 1: To find a basis for the space of 2×2 lower triangular matrices, we need to determine a set of linearly independent matrices that span this space.
A 2×2 lower triangular matrix has the form:

[\begin{matrix} a & 0 \\ c & b \end{matrix}]

where a, b, and c are elements of a field, such as real numbers or complex numbers.
Let's denote the matrix above as matrix A. To find a basis, we need to consider different values of a, b, and c that give distinct matrices. We'll start with a = 1, b = 0, and c = 0:

A_{1} = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]

Step 2: Next, we'll consider a = 0, b = 1, and c = 0:

A_{2} = [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]

Lastly, we'll consider a = 0, b = 0, and c = 1:

A_{3} = [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}]

These three matrices,

A_{1}

,

A_{2}

, and

A_{3}

, are linearly independent and span the space of 2×2 lower triangular matrices. Therefore, they form a basis for this space.
The basis is given by:

Basis = {A_{1}, A_{2}, A_{3}}

Alternatively, we can express the basis as a set of column vectors:

Basis = {[\begin{matrix} 1 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \end{matrix}]}

Each of these column vectors represents one of the matrices

A_{1}

,

A_{2}

, and

A_{3}

.

alenahelenash · Accepted Answer

[a0bc]To form a basis, we need linearly independent vectors that span the space of lower triangular matrices. We can choose the following matrices as a basis:𝐯1=[1000],𝐯2=[0010],𝐯3=[0100].Therefore, the basis for the space of 2×2 lower triangular matrices is given by:Basis={𝐯1,𝐯2,𝐯3}.

Find a basis for the space of 2 \times 2

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