Vectors: linear transformation and bases question

In summary: Therefore, there does not exist any linear transformation S: W → V such that S ◦ T = I, and T cannot exist as a linear transformation.In summary, the conversation discusses proving that a set is a basis for a linear transformation from V to W, and showing that there does not exist any linear transformation from W to V that can be composed with T to result in the identity transformation. The first part involves proving that the set {(1, 0), (0, 1)} is a basis for V, while the second part involves showing that the matrices N(T) and R(T) do not have a
  • #1
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Homework Statement


Homework Equations



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The Attempt at a Solution



i have answered the first part, explained why its a basis

i think that i need to show that the N(T) = V alongside R(T) = W will give matrices that cannot be multiplied therefore T is the one that can't exist
not sure though

and i think for the final part i just multiply such matrix but i don't know what to multiply?!

very confusing!
 
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  • #2
Let V = R2, W = R3 and T:V → W be a linear transformation defined byT(x,y) = (x + y, 2x + y, x - y). Part 1: Prove that {(1, 0), (0, 1)} is a basis for V.To show that {(1, 0), (0, 1)} is a basis for V, we must show that it spans V and is linearly independent. To show that it spans V, we need to show that all elements in V can be written as linear combinations of (1, 0) and (0, 1). Let v = (a, b) be an arbitrary element of V. Then v = a(1, 0) + b(0, 1) = (a, 0) + (0, b) = (a + 0, 0 + b) = (a, b).This shows that any element of V can be written as a linear combination of (1, 0) and (0, 1), so the set {(1, 0), (0, 1)} spans V.To show that the set is linearly independent, we need to show that the only solution to the equation a(1, 0) + b(0, 1) = 0 is when a = b = 0. Substituting a = 0 and b = 0 into the equation gives:0(1, 0) + 0(0, 1) = 0 which is true, so the only solution is when a = b = 0. This shows that the set is linearly independent, so {(1, 0), (0, 1)} is a basis for V.Part 2: Show that there does not exist any linear transformation S: W → V such that S ◦ T = I.To show that there does not exist any linear transformation S: W → V such that S ◦ T = I, we need to show that the matrices N(T) and R(T) do not have a common multiple. Let N(T) = [1 1 2] and R(T) = [1 2 -1]. These matrices cannot be multiplied together because their number of columns (
 

FAQ: Vectors: linear transformation and bases question

1. What is a vector in linear algebra?

A vector in linear algebra is a mathematical object that represents both magnitude and direction. It is typically represented graphically as an arrow pointing in a specific direction with a specific length.

2. What is a linear transformation?

A linear transformation is a function that maps one vector space to another, while preserving the operations of vector addition and scalar multiplication. In other words, the image of any linear combination of vectors is equal to the same linear combination of the images of those vectors.

3. How do you determine if a transformation is linear?

A transformation is linear if it satisfies two properties: additivity (T(u+v) = T(u) + T(v)) and homogeneity (T(cu) = cT(u)) for all vectors u and v and scalar c. In other words, the transformation must preserve vector addition and scalar multiplication.

4. What is a basis in linear algebra?

A basis 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 unique linear combination of the basis vectors.

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

To find the basis of a vector space, you can start by finding a set of linearly independent vectors that span the space. Then, if necessary, you can use the Gram-Schmidt process to orthogonalize the vectors and create an orthonormal basis. Alternatively, you can use the row reduction method to find the pivot columns in the matrix representing the vector space, which will give you the basis vectors.

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