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Bob
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I think R2 is a subspace of R3 in the form(a,b,0)'.
To begin with, for W to be a subspace of V, it must be a subset of V. Things in R^2 are of the form (a, b), with two components while things in R^3 are of the form (a, b, c) with three components. Members of R^2 are not members of R^3 so R^2 is not a subset of R^3.MaxManus said:I know that it is an old thread, but I still don't get why R^2 is not a subspace of R^3. Is it only because R^3 has 3 components and R^2 only 2 components? Is it possible to use the three conditions to show that R^2 is not a subspace of R^3?
1. The zero vector, 0, is in W.
2. If u and v are elements of W, then the sum u + v is an element of W;
3. If u is an element of W and c is a scalar from K, then the scalar product cu is an element of W;
R2 is not a subspace of R3 because the two spaces have different dimensions. R2 is a two-dimensional vector space, while R3 is a three-dimensional vector space. This means that the two spaces have different bases and therefore, vectors in R2 cannot be represented in R3 and vice versa.
No, a subset of R3 cannot be a subspace of R2. A subspace must satisfy the three conditions of closure under addition, closure under scalar multiplication, and contain the zero vector. Since R2 and R3 have different dimensions, a subset of R3 cannot satisfy these conditions and therefore, cannot be a subspace of R2.
The main difference between R2 and R3 is their dimensions. R2 is a two-dimensional vector space, while R3 is a three-dimensional vector space. This means that R3 has one more dimension than R2, and therefore, can represent more complex and diverse mathematical concepts.
No, a subspace of R3 cannot be a subspace of R2. This is because a subspace must satisfy the three conditions of closure under addition, closure under scalar multiplication, and contain the zero vector. Since R2 and R3 have different dimensions, a subspace of R3 cannot satisfy these conditions and therefore, cannot be a subspace of R2.
The number of dimensions affects vector spaces by determining the maximum number of independent vectors that can exist in that space. For example, R2 can have a maximum of two independent vectors, while R3 can have a maximum of three independent vectors. This also affects the types of operations and transformations that can be performed in each space.