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Questions and Answers
Does a polynomial of degree less than 7 form a 7-dimensional subspace of the linear space of all polynomials?
Does a polynomial of degree less than 7 form a 7-dimensional subspace of the linear space of all polynomials?
True (A)
The claim is that the function T(f)=3f-4f' is a linear transformation.
The claim is that the function T(f)=3f-4f' is a linear transformation.
True (A)
The claim is that all 2x2 triangular matrices form a subspace of the space of all 2x2 matrices.
The claim is that all 2x2 triangular matrices form a subspace of the space of all 2x2 matrices.
True (A)
The claim is that the kernel of a linear transformation is a subspace of the domain.
The claim is that the kernel of a linear transformation is a subspace of the domain.
The claim is that the space R^(2x3) is 5-dimensional.
The claim is that the space R^(2x3) is 5-dimensional.
The claim is that if f1,...,f(n) is a basis of a linear space V, then any element of V can be written as a linear combination of f1,...,f(n).
The claim is that if f1,...,f(n) is a basis of a linear space V, then any element of V can be written as a linear combination of f1,...,f(n).
The claim is that the space P1 is isomorphic to C.
The claim is that the space P1 is isomorphic to C.
The claim is that if the kernel of a linear transformation T from P4 to P4 is {0}, then T must be an isomorphism.
The claim is that if the kernel of a linear transformation T from P4 to P4 is {0}, then T must be an isomorphism.
The claim is that if W1 and W2 are subspaces of a linear space V, then the intersection W1 ∩ W2 must be a subspace of V as well.
The claim is that if W1 and W2 are subspaces of a linear space V, then the intersection W1 ∩ W2 must be a subspace of V as well.
The claim is that all bases of P3 contain at least one polynomial of degree equal to or less than 2.
The claim is that all bases of P3 contain at least one polynomial of degree equal to or less than 2.
The claim is that if T is an isomorphism then T(-1) is an isomorphism as well.
The claim is that if T is an isomorphism then T(-1) is an isomorphism as well.
The claim is that all linear transformations from P3 to R^(2x2) are isomorphisms.
The claim is that all linear transformations from P3 to R^(2x2) are isomorphisms.
The claim is that the space of all upper triangular 4x4 matrices is isomorphic to the space of all lower triangular 4x4 matrices.
The claim is that the space of all upper triangular 4x4 matrices is isomorphic to the space of all lower triangular 4x4 matrices.
The claim is that there exists a 2x2 matrix A such that the space of all matrices commuting with A is one-dimensional.
The claim is that there exists a 2x2 matrix A such that the space of all matrices commuting with A is one-dimensional.
The claim is that if W is a subspace of V and W is finite dimensional then V must be finite dimensional as well.
The claim is that if W is a subspace of V and W is finite dimensional then V must be finite dimensional as well.
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Study Notes
Polynomials and Subspaces
- A polynomial of degree less than 7 forms a 7-dimensional subspace, with a basis including constants and terms up to (t^6).
- Subspaces have closed operations under addition and scalar multiplication, exemplified by lower triangular matrices.
Linear Transformations
- The function (T(f)=3f-4f') qualifies as a linear transformation since both components are linear.
- The kernel of a linear transformation indicates that if it's {0}, then the transformation is an isomorphism.
Matrix Spaces and Dimensions
- (R^{2x3}) is 6-dimensional, accommodating all possible 2x3 matrices, contrary to the claim that it is 5-dimensional.
- The dimension of (P_1) is 1 while the dimension of (\mathbb{C}) is 2, establishing that they are not isomorphic.
Isomorphism Criteria
- A transformation is deemed an isomorphism if its kernel is zero, and its image before and after transformation remains identical.
- If (T) is an isomorphism, then its inverse (T^{-1}) remains an isomorphism due to linearity.
Subspace Intersections
- The intersection of two subspaces (W_1) and (W_2) yields another subspace, consistent with the properties of subspace definitions.
Basis Requirements
- A basis of (P_3) must include polynomials of degrees 0, 1, 2, and 3 to adequately span the space.
- There cannot be a 2x2 matrix (A) such that the space of matrices commuting with (A) is one-dimensional, as (I_2) commutes with any 2x2 matrix.
Dimensional Relationships
- A finite-dimensional subspace (W) does not guarantee that the entire space (V) is finite-dimensional.
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