Does every linear transformation from a complex vector space to itself have an eigenvector?

Expert Solution

Colby Messinger answered 1 year ago

let l be the linear transformation from a vector space V where ker(L)=0 if { v1,v2,v3}...

let l be the linear transformation from a vector space V where ker(L)=0 if { v1,v2,v3} are linearly independent vectors on V prove {Lv1,Lv2,Lv3} are linearly independent vectors in V

Let T be a linear operator on a finite-dimensional complex vector space V . Prove that...

Let T be a linear operator on a finite-dimensional complex vector space V . Prove that T is diagonalizable if and only if for every λ ∈ C, we have N(T − λIV ) = N((T − λIV )2).

Let ? and W be finite dimensional vector spaces and let ?:?→? be a linear transformation....

Let ? and W be finite dimensional vector spaces and let ?:?→? be a linear transformation. We say a linear transformation ?:?→? is a left inverse of ? if ST=I_v, where ?_v denotes the identity transformation on ?. We say a linear transformation ?:?→? is a right inverse of ? if ??=?_w, where ?_w denotes the identity transformation on ?. Finally, we say a linear transformation ?:?→? is an inverse of ? if it is both a left and right...

Find a matrix representation of transformation T(x)= 2x1w1+x2w2-3x3w3 from R3 to a vector space W, where...

Find a matrix representation of transformation T(x)= 2x1w1+x2w2-3x3w3 from R3 to a vector space W, where w1,w2, and w3 ∈ W. Clearly state how this matrix is representing the transformation.

(a) Find a linear transformation T : R2→R2 that (i) maps the x1-axis to itself, (ii)...

(a) Find a linear transformation T : R2→R2 that (i) maps the x1-axis to itself, (ii) maps the x2-axis to itself, and (iii) maps no other line through the origin to itself. For example, the negating function (n: R2→R2 defined by n(x) =−x) satisfies (i) and (ii), but not (iii). (b) The function that maps (x1, x2) to the perimeter of a rectangle with side lengths x1 and x2 is not a linear function. Why? For part (b) I can't...

For a linear transformation between two finite-dimensional vector spaces. A) State the "Rank-Nully Theorem" for the...

For a linear transformation between two finite-dimensional vector spaces. A) State the "Rank-Nully Theorem" for the linear transformation. B) Prove the "Rank-Nully Theorem" you just stated in (A).

Let T be an operator on a finite-dimensional complex vector space V, and suppose that dim...

Let T be an operator on a finite-dimensional complex vector space V, and suppose that dim Null T = 3, dimNullT2 =6. Prove that T does not have a square root; i.e. there does not exist any S ∈ L (V) such that S2 = T.

suppose that T : V → V is a linear map on a finite-dimensional vector space...

suppose that T : V → V is a linear map on a finite-dimensional vector space V such that dim range T = dim range T2. Show that V = range T ⊕null T. (Hint: Show that null T = null T2, null T ∩ range T = {0}, and apply the fundamental theorem of linear maps.)

Linear Algebra: Explain what a vector space is and offer an example that contains at least...

Linear Algebra: Explain what a vector space is and offer an example that contains at least five (5) of the ten (10) axioms for vector spaces.

(3) (a) Show that every two-dimensional subspace of R3 is the kernel of some linear transformation...

(3) (a) Show that every two-dimensional subspace of R3 is the kernel of some linear transformation T : R3 → R. [Hint: there are many possible ways to approach this problem. One is to use the following fact, typically introduced in multivariable calculus: for every plane P in R3, there are real numbers a, b, c, d such that a point (x,y,z) belongs to P if and only if it satisfies the equation ax+by+cz = d. You may use this...

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