Linear subspaces challenging question

In summary, the question asks whether the set S is a subspace of the vectorspace V, where V is represented by Rn and S is the solution set of the system Ax=b, where A is an mxn matrix. The key to proving this is to show that S satisfies the properties of a subspace, namely closure under addition and scalar multiplication.
  • #1
aan
1
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Please anyone solve this question or can even email me on my ID abu_95bakar@yahoo.com...

For the following question determine whether the set S is a sub space of the given vectorspace V.

v=Rn( where n represent dimension), S is the solution set of the sysytem Ax=b, where A is an mxn matrix.



PLEASE HELP! THANX A MILLION IN ADVANCE.
 
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  • #2
aan said:
Please anyone solve this question or can even email me on my ID abu_95bakar@yahoo.com...

For the following question determine whether the set S is a sub space of the given vectorspace V.

v=Rn( where n represent dimension), S is the solution set of the sysytem Ax=b, where A is an mxn matrix.



PLEASE HELP! THANX A MILLION IN ADVANCE.
Looks straight forward to me. To prove something is a subspace, show that it satisifies the properties of a subspace: specifically that it is closed under addition and scalar multiplication. If x and y are in this set then Ax= b and Ay= b so A(x+y)= Ax+ Ay= b+ b= 2b. What does that tell you?
 

1. What is a linear subspace?

A linear subspace is a subset of a vector space that is closed under addition and scalar multiplication. In other words, it contains all linear combinations of its elements.

2. How is a linear subspace different from a vector space?

A vector space is a set of vectors that can be added and multiplied by scalars, while a linear subspace is a subset of a vector space with the same properties.

3. What are some examples of linear subspaces?

Some examples of linear subspaces include the x-y plane in three-dimensional space, the set of all polynomials of degree n or less, and the null space of a matrix.

4. What makes linear subspaces challenging to study?

Linear subspaces can be challenging to study because they can have infinite dimensions and can be difficult to visualize. Additionally, understanding the properties of subspaces requires a strong understanding of linear algebra.

5. How are linear subspaces used in real-world applications?

Linear subspaces have many practical applications, such as in image and signal processing, data compression, and machine learning. They are also used in physics and engineering to model and solve complex systems.

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