1 Let V Be A Finite Dimensional Vector Space Suppose That S Is A Subset Of V Which Is Linearly Independent Then S 1 (54.3 KiB) Viewed 34 times
1 Let V Be A Finite Dimensional Vector Space Suppose That S Is A Subset Of V Which Is Linearly Independent Then S 2 (59.22 KiB) Viewed 34 times
1. Let V be a finite dimensional vector space. Suppose that S is a subset of V which is linearly independent. Then |S| ≤ dim(V). 2. Let V be a vector space. Let V₁, V2 € V. Suppose that there exists a vector v € V such that v+v₁ = V2 + v. Then V1 = V2. 3. {(a, 1): a € R} is a subspace of R². 4. Let V be a vector space over a field F. Let S = {V1, V2, V3} V. Suppose that S is linearly dependent and Oy S. Then there & exists a1, a2 € F such that a₁ v₁ + a₂ · V2 = V3. 5. Let V and W be finite dimensional vector spaces. Let TV → W be a linear transformation. Then there exists a linear transformation U: W → V such that U o T = Iv. 6. Let V be a vector space and S be a non- empty subset of V. Then span (span (S)) = span(S).
7. Let V and W be vector spaces and T, U: V → W be linear transformations. Then Z:= {ve V: T(v) = U(v)} is a subspace of V. 8. Let S = {V1, V2, V3, V4, V5} C P₂ (R) such that span (S) = P₂(R). Then there exists a subset of S which is a basis for P₂ (R). 9. There exists a vector space V and a linear transformation T: V → V such that T is surjective but not injective. 10. Let V and W be finite dimensional vector spaces, B = {v₁, ..., Vn} be a basis for V, and T V W be an injective linear transformation. Then {T(v₁), ..., T(vn)} is a basis for W.
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