Finding basis of spaces and dimension

In summary, for the given space of 2x2 matrices in R^2x2 where a+d=0, a possible combination of basis is [1 0, 0 0], [0 1, 0 0], and [0 0, 1 0]. The dimension of the space is 3. To find a basis, one needs to come up with elements that can be written as a linear combination of the basis vectors. The terms "elements" and "linear sum" refer to the vectors in the basis set and the vector addition with scalar multiplication, respectively.
  • #1
maximade
27
0

Homework Statement


Find a basis for each of the spaces and determine its dimension:
The space of all matrices A=[a b, c d] (2x2 matrix) in R^(2x2) such that a+d=0


Homework Equations





The Attempt at a Solution


So I jumped at this question without knowing too much about spaces and dimensions, but:
I think a possible combination of basis can be: [1 0, 0 0]. [0 1, 0 0]. [0 0, 1 0] (not sure if [0 0, 0 -1] would be considered since d would be negative in this case) Also from that I assume the dimension is 3?
Truthfully even if I got it right, I do not even know what happened. Can someone conceptually tell me what I am doing exactly?
Thanks in advance.
 
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  • #2
you need to come up with a basis, that has elements, such that any 2x2 matrix with a=d can be written as a linear sum of the basis elements.

clearly the basis elements will need to satisfy being a 2x2 matrix with d=a, you first element does not
 
  • #3
note this is entirely equivalent to considering 4 component vectors in R^4, with x_1 = x_4

often the vector form is easier to conceptualise
 
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  • #4
lanedance said:
you need to come up with a basis, that has elements, such that any 2x2 matrix with a=d can be written as a linear sum of the basis elements.

clearly the basis elements will need to satisfy being a 2x2 matrix with d=a, you first element does not

Can you explain to me how d=a, since a+d=0?
Also since I am actually having a hard time learning this on my own, can you tell me what you mean by "elements" and "linear sum"?
 
  • #5
maximade said:
Can you explain to me how d=a, since a+d=0?
Also since I am actually having a hard time learning this on my own, can you tell me what you mean by "elements" and "linear sum"?
good pickup, should be a = -d

if you're not familair with those terms, you may need to do a bit of reading.. though i have been a little loose with terminiology

in post #4 i actually meant component and have changed accordingly

a basis, is a set of vectors that spans a vector space

an element is a member of a set, for example a vector in the basis set

a linear combination (or sum) is a vector addition with scalar multiplication
eg. if u,v are vectors, and a,b are scalars, then
w = au + bv is a linear combination

a set, S, of vectors spans a space if any vector in the space can be written as a linear combination of vectors in S
 
  • #6
From a+ d= 0 you get d=-a as you say. You can then write
[tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}= \begin{bmatrix}a & b \\ c & -a\end{bmatrix}[/tex]
[tex]= \begin{bmatrix}a & 0 \\ 0 & -a\end{bmatrix}+ \begin{bmatrix}0 & b \\ 0 & 0\end{bmatrix}+ \begin{bmatrix}0 & 0 \\ c & 0\end{bmatrix}[/tex]
and the dimension and a basis should be clear.
 

1. What is the definition of a basis in linear algebra?

A basis in linear algebra is a set of linearly independent vectors that span a vector space. This means that any vector in the vector space can be written as a linear combination of the basis vectors. A basis is also the smallest set of vectors that can span the vector space.

2. How do you find the basis of a vector space?

To find the basis of a vector space, you need to first find a set of linearly independent vectors that span the vector space. This can be done by solving a system of linear equations or by using the row reduction method. Once you have a set of linearly independent vectors, you can check if they span the vector space by seeing if any vector in the space can be written as a linear combination of the basis vectors.

3. What is the dimension of a vector space?

The dimension of a vector space is the number of vectors in its basis. It represents the minimum number of vectors needed to span the entire vector space. This is also equal to the number of rows or columns in a matrix that represents the vector space.

4. Can a vector space have more than one basis?

Yes, a vector space can have more than one basis. This is because there can be multiple sets of linearly independent vectors that span the same vector space. However, all bases for a given vector space will have the same number of vectors, which is the dimension of the vector space.

5. How does the basis of a subspace relate to the basis of the original vector space?

The basis of a subspace is a subset of the basis of the original vector space. This is because a subspace is a smaller vector space within the larger vector space. The basis of the subspace will have fewer vectors than the basis of the original space, but they will still span the subspace and be linearly independent.

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