Recent content by cocobaby

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    Simple question on non singular linear transformation

    Homework Statement Given that "If T(Ta)=0, then Ta=0", can we say that the linear transformation on V is nonsingular? Homework Equations The Attempt at a Solution Since what the statement implies is that T has only zero subspace of V as its null space, can we not say that it's...
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    Is There a General Relationship Between the Range of T and the Null Space of T?

    Given a linear transformation T from V to V, can we say that the range of T is in the space spanned by the column vectors of T. And we already know that the null space of T is the one spanned by the set of vectors that are orthogonal to the row vectors of T, then is there any general...
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    Isomorphism and Subspace Intersection in Complex Vector Space

    Homework Statement Let V be a vector space over the field of complex numbers, and suppose there is an isomorphism T of V onto C3. Let a1, a2, a3,a4 be vectors in V such that Ta1 = (1, 0 ,i) Ta2 = (-2, 1+i, 0) Ta3 = (-1, 1, 1) Ta4 = (2^1/2, i, 3) Let W1 be the suubspace spanned by a1...
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    Solving Complex Linear Problems on Vector Spaces

    Homework Statement Let V be the set of all complex numbers regarded as a vector space over the field of real numvers. Find a function from V into V which is a linear transformation on the above vector space, but which is not a linear transformation on C1, i.e.,which is not complex linear...
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    Do Subsets of Natural Numbers Have Cluster Points?

    Does any subset of natural number N have a cluster point? And does it diverge?
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    Proving BA=I using Elementary Row Operations and Determinants

    Let A and B be 2x2 matrices s.t. AB=I . Then how can I prove that BA=I? I assumed that there must exist some sequence of elementary row operations which carries B into I, and I denoted this sequence by the matrix A. But here, I realized there's some pieces that I' m missing, which I...
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    Using the method of mathematical induction

    -------------------------------------------------------------------------------- 1. Homework Statement Let H be a ten- element set of potive integers ranged from 1 to 99. Prove that H has two disjoint subsets A and B so that the sum of the elements of A is equal to the sum of the elements of B.
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    Using the mathod of mathematical induction

    Homework Statement Let H be a ten- element set of potive integers ranged from 1 to 99. Prove that H has two disjoint subsets A and B so that the sum of the elements of A is equal to the sum of the elements of B.
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    Undergrad Subfield of the field of complex numbers

    Can anyone prove to me why each subfield of the field of complex numbers contains every rational numers?