Linear algebra problem related to vector subspace

The bases are both empty sets, as the dimensions of ker f and I am f are both 0.5. No, since f is not injective, i.e. there exist two different inputs that give the same output. For example, f(1,1,0,3)=f(2,1,1,6)=(1,1,0,3).6. A diagonal matrix for f is not unique. One possible diagonal matrix is D=diag(1,1,0,3).
  • #1
Montgomery
2
0

Homework Statement


X ={(x1,x2,x2 −x1,3x2):x1,x2 ∈R}
f(x1,x2,x2 −x1,3x2)=(x1,x1,0,3x1)
1. Find a basis for X.
2. Find dim X.
3. Find ker f and I am f
4. Find bases for ker f and I am f
5. Is f a bijection? Why?
6. Find a diagonal matrix for f.

Homework Equations

The Attempt at a Solution


1. Put x1=x2=1: (1, 1, 0, 3) and Put x1=1 and x2=2: (1, 2, 1, 6)
2. Dim X = 2 as there are two vectors
3. Ker f = 0, I am f = X
4. (0,0,0,0)
5. I guess no, but do not know how to explain
6. No idea
 
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  • #2
1. Looks like a good guess, but you will need to prove that the two vectors you found span X.

2. Here you will also need to prove that the two vectors you found in 1 are linearly independent.

3. ker f is a set, not a vector or a number. If you meant ker f={0}, where 0 is the zero vector, then the answer is wrong since e.g. f(0,1,1,3)=0. Your answer for I am f is wrong too, since (1,2,1,6) is in X but not in I am f.
 

1. What is a vector subspace?

A vector subspace is a subset of a vector space that satisfies the properties of a vector space. This means that it is closed under addition and scalar multiplication, and contains the zero vector.

2. What is the difference between a vector subspace and a vector space?

A vector subspace is a subset of a vector space, while a vector space is a set of vectors that can be added together and multiplied by scalars. A vector subspace must contain the zero vector and be closed under addition and scalar multiplication, while a vector space can have any number of vectors and does not necessarily have to satisfy these properties.

3. How can I determine if a set of vectors forms a vector subspace?

To determine if a set of vectors forms a vector subspace, you can check if they satisfy the properties of a vector space. This means that they must be closed under addition and scalar multiplication, and contain the zero vector. You can also check if they are linearly independent, meaning that none of the vectors can be written as a linear combination of the others.

4. Can a vector subspace contain an infinite number of vectors?

Yes, a vector subspace can contain an infinite number of vectors. As long as it satisfies the properties of a vector space, it can contain any number of vectors. However, it must always contain the zero vector.

5. How is linear algebra used to solve problems related to vector subspaces?

Linear algebra is used to solve problems related to vector subspaces by providing a framework for understanding and manipulating vectors and vector spaces. It allows us to represent and solve systems of linear equations, determine if a set of vectors forms a subspace, and find bases and dimensions of subspaces. This is useful in many scientific fields, including physics, engineering, and computer science.

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