Problem 1E: The polynomials of degree less than 7 form a seven dimensional subspace of the linear space of all... Problem 2E Problem 3E Problem 4E Problem 5E: The space 23 is five-dimensional. Problem 6E Problem 7E Problem 8E Problem 9E: If W1 and W2 are subspaces of a linear space V, then the intersection W1W2 must be a subspace of V... Problem 10E: If T is a linear transformation from P6 to 22 , then the kernel of T must be three-dimensional. Problem 11E Problem 12E Problem 13E Problem 14E: All linear transformations from P3 to 22 are isomorphisms. Problem 15E: If T is a linear transformation from V to V, then the intersection of im(T) and ker(T) must be {0} . Problem 16E Problem 17E: Every polynomial of degree 3 can be expressed as a linear combination of the polynomial (t3),(t3)2 ,... Problem 18E: a linear space V can be spanned by 10 elements, then the dimension of V must be 10 . Problem 19E Problem 20E: There exists a 22 matrix A such that the space V of all matrices commuting with A is... Problem 21E Problem 22E Problem 23E Problem 24E Problem 25E Problem 26E Problem 27E Problem 28E Problem 29E Problem 30E Problem 31E: If W is a subspace of V, and if W is finite dimensional, then V must be finite dimensional as well. Problem 32E Problem 33E Problem 34E Problem 35E Problem 36E Problem 37E Problem 38E Problem 39E Problem 40E Problem 41E Problem 42E: The transformation D(f)=f from C to C is an isomorphism. Problem 43E: If T is a linear transformation from P4 to W with im(T)=W , then the inequality dim(W)5 must hold. Problem 44E: The kernel of the linear transformation T(f(t))=01f(t)dt from P to is finite dimensional. Problem 45E: If T is a linear transformation from V to V, then {finV:T(f)=f} must be a subspace of V. Problem 46E: If T is a linear transformation from P6 to P6 that transforms tk into a polynomial of degree k (for... Problem 47E: There exist invertible 22 matrices P and Q such that the linear transformation T(M)=PMMQ is an... Problem 48E: There exists a linear transformation from P6 to whose kernel is isomorphic to 22 . Problem 49E: If f1,f2,f3 is a basis of a linear space V, and if f is any element of V, then the elements... Problem 50E: There exists a two-dimensional subspace of 22 whose nonzero elements are all invertible. Problem 51E: The space P11 is isomorphic to 34 . Problem 52E: If T is a linear transformation from V to W, and if both im(T) and ker(T) are finite dimensional,... Problem 53E: If T is a linear transformation from V to 22 with ker(T)={0} , then the inequality dim(V)4 must... Problem 54E: The function T(f(t))=ddt23t+4f(x)dx from P5 to P5 is an isomorphism. Problem 55E: Any four-dimensional linear space has infinitely many three-dimensional subspaces. Problem 56E: If the matrix of a linear transformation T (with respect to some basis) [3504] , then there must... Problem 57E: If the image of a linear transformation T is infinite dimensional, then the domain of T must be... Problem 58E: There exists a 22 matrix A such that the space of all matrices commuting with A is... Problem 59E: If A, B, C, and D are noninvertible 22 matrices, then the matrices AB, AC, and AD must be linearly... Problem 60E: There exist two distinct three-dimensional subspaces W1 and W2 of P4 such that the union W1W2 is a... Problem 61E: the elements f1,...,fn , (where f10 ) are linearly dependent, then one element fk can be expressed... Problem 62E: There exists a 33 matrix P such that the linear transformation T(M)=MPPM from 33 to 33 is an... Problem 63E: If f1,f2,f3,f4,f5 are elements of a linear space V, and if there are exactly two redundant elements... Problem 64E: There exists a linear transformation T from P6 to P6 such that the kernel of T is isomorphic to the... Problem 65E: If T is a linear transformation from V to W, and if both im(T) and ker(T) are finite dimensional,... Problem 66E: If the matrix of a linear transformation T (with respect to some basis) is [3504] then there must... Problem 67E: Every three-dimensional subspace of 22 contains at least one invertible matrix. format_list_bulleted