Find a basis for the space of 2 times 2 diagonal matrices. text{Basis }=left{begin{bmatrix}&& end{bmatrix},begin{bmatrix}&& end{bmatrix}right}

Chardonnay Felix

Chardonnay Felix

Answered question

2021-01-31

Find a basis for the space of 2×2 diagonal matrices.
Basis ={[],[]}

Answer & Explanation

Jayden-James Duffy

Jayden-James Duffy

Skilled2021-02-01Added 91 answers

Step 1
let the space of 2×2 diagonal matrices be as
[a00b]
we can choose values of a and b randomly Step 2
Basis of matrix [a00b] can be taken as
a=1,b=0
[1000]
a=0, b=1 [0001]
Basis ={[1000],[0001]}
Don Sumner

Don Sumner

Skilled2023-05-09Added 184 answers

We need to identify a collection of linearly independent matrices that cover the space of 2*2 diagonal matrices in order to construct a basis for the space.
D=[d100d2]
where d1 and d2 are arbitrary constants.
To form a basis, we need to find distinct diagonal matrices that are linearly independent. Here's a possible basis for the space of 2×2 diagonal matrices:
B={[1000],[0001]}
We have two distinct diagonal matrices, and we can see that no linear combination of one matrix can produce the other. Therefore, these matrices are linearly independent.
To confirm that these matrices span the space, we can take an arbitrary 2×2 diagonal matrix:
D=[d100d2]
We can rewrite this matrix as a linear combination of the basis matrices:
D=d1[1000]+d2[0001]
This shows that any 2×2 diagonal matrix can be expressed as a linear combination of the basis matrices. Therefore, the set B is a basis for the space of 2×2 diagonal matrices.
karton

karton

Expert2023-05-09Added 613 answers

To find a basis for the space of 2×2 diagonal matrices, we can start by considering the general form of a diagonal matrix:
[a00b]
where a and b are scalars.
We can see that any 2×2 diagonal matrix can be written as a linear combination of the following matrices:
[1000],[0001],[0100]
Let's call these matrices D1, D2, and D3, respectively. We can express any 2×2 diagonal matrix as:
aD1+bD2+cD3
where a, b, and c are scalars.
Therefore, the set {D1,D2,D3} forms a basis for the space of 2×2 diagonal matrices.
Alternatively, we can express the basis as:
{[1000],[0001],[0100]}
user_27qwe

user_27qwe

Skilled2023-05-09Added 375 answers

A set of matrices that span the space and are linearly independent must be found in order to establish a basis for the space of 2x2 diagonal matrices.
A 2x2 diagonal matrix can be represented as:
[a00b]
where a and b are any real numbers.
To find a basis, we can consider two diagonal matrices with different values for a and b. A possible basis for the space of 2x2 diagonal matrices can be:
D1=[1000],D2=[0001].
These two matrices span the space of 2x2 diagonal matrices since any diagonal matrix can be written as a linear combination of D1 and D2.
To show that the basis is linearly independent, we need to verify that the only solution to the equation c1D1+c2D2=0 (where 0 represents the zero matrix) is c1=c2=0.
[c1000]+[000c2]=[0000]
This implies c1=c2=0. Hence, the basis B={D1,D2} is linearly independent.
Therefore, the basis for the space of 2x2 diagonal matrices is given by:
B={[1000],[0001]}.

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