Template problems in linear algebra

Let T:R3R3 be the linear transformation that rotates counterclockwise around the z-axis by 2π3.

  1. Write the matrix for T with respect to the standard basis {[100],[010],[001]}.
  2. Write the matrix for T with respect to the basis {[32120],[010],[001]}.
  3. Determine all (complex) eigenvalues of T.
  4. Is T diagonalizable over C? Justify your answer.

Let V denote the real vector space of polynomials in x of degree at most 3. Let B={1,x,x2,x3} be a basis for V and T:VV be the function defined by T(f(x))=f(x)+f(x).

  1. Prove that T is a linear transformation.
  2. Find [T]B, the matrix representation for T in terms of the basis B.
  3. Is T diagonalizable? If yes, find a matrix A so that A[T]BA1 is diagonal, otherwise explain why T is not diagonalizable.

Let Mn(R) be the vector space of all n×n matrices with real entries. We say that A,BMn(R) commute if AB=BA.

  1. Fix AMn(R). Prove that the set of all matrices in Mn(R) that commute with A is a subspace of Mn(R).
  2. Let A=[1111]M2(R) and let WM2(R) be the subspace of all matrices of M2(R) that commute with A. Find a basis of W.

Let VR5 be the subspace defined by the equation

x12x2+3x34x4+5x5=0.

  1. Find (with justification) a basis for V.
  2. Find (with justification) a basis for V, the subspace of R5 orthogonal to V under the usual dot product.

Let T:R3R3 be the linear transformation defined by T([xyz])=[x+y2zxy+2z].

  1. Find the matrix that represents T with respect to the standard basis for R3.
  2. Find a basis for the kernel of T.
  3. Determine the rank of T.

Let A=[002121103].

  1. Determine whether A is diagonalizable, and if so, give its diagonal form along with a diagonalizing matrix.
  2. Compute A42. Remember to show all work.

Let A=[211101110].

  1. Compute the characteristic polynomial pA(x) of A. It has integer roots.
  2. For each eigenvalue λ of A, find a basis for the eigenspace Eλ.
  3. Determine if A is diagonalizable. If so, give matrices P and B such that P1AP=B and B is diagonal. If no, explain carefully why A is not diagonalizable.

Let A=[6211032001].

  1. Find bases for the eigenspaces of A.
  2. Determine if A is diagonalizable. If so, give an invertible matrix P and diagonal matrix D such that P1AP=D. If not, explain why not.

Let WR5 be the subspace spanned by the set of vectors {1,2,0,2,1,2,4,1,1,2,0,1,2,2,1}.

  1. Compute the dimension of W.
  2. Determine the dimension of W, the perpendicular subspace in R5.
  3. Find a basis for W.

Consider the following matrix:

A=[0100001000040000].

  1. Determine the characteristic and minimal polynomials of A.
  2. Find a basis for R4 consisting of generalized eigenvectors of A.
  3. Find an invertible matrix S such that S1AS is in Jordan canonical form.
  4. Determine a Jordan canonical form of A.

Let A=[211522733].

  1. Find the characteristic polynomial and the minimal polynomial of A.
  2. Find the Jordan canonical form of the matrix A.

Let P3 be the real vector space of all real polynomials of degree three or less. Define L:P3P3 by L(p(x))=p(x)+p(x).

  1. Prove L is a linear transformation.
  2. Find a basis for the null space of L.
  3. Compute the dimension of the image of L.

Let V={a0+a123+a243a0,a1,a2Q}R. This set is a vector space over Q.

  1. Verify V is closed under product (using the usual product operation in R).
  2. Let T:VV be the linear transformation defined by T(v)=(23+43)v. Find the matrix that represents T with respect to the basis {1,23,43} for V.
  3. Determine the characteristic polynomial for T.

Suppose {v1,v2,v3} is a basis for R3 and T:R3R3 is a linear transformation satisfying the following:

T(v1)=4v1+2v2T(v2)=5v2T(v3)=2v1+4v2+5v3.

Determine the eigenvalues of T and find a basis for each eigenspace.


LetWR5 be the space spanned by the vectors

{[12021],[24112],[01221]}.

  1. Compute the dimension of W.
  2. Let W={vR5vw=0 for all wW}. Determine the dimension of W, and explain how this following immediately from (a) using a theorem.
  3. Find a basis for W.

Let L be the line in R2 defined by y=3x, and let T:R2R2 be the linear transformation that orthogonally projects onto L and then stretches along L by a factor of two.

  1. Find the eigenvalues and an eigenbasis B for T.
  2. Determine the matrix for T with respect to the basis B.
  3. Determine the matrix for T with respect to the standard basis.

Let T:R3R3 be the orthogonal projection to a 1-dimensional linear subspace LR3.

  1. List the eigenvalues of T.
  2. Write the characteristic polynomial pT(x) for T.
  3. Is T diagonalizable? Briefly justify your answer.

Let T:R3R3 be the orthogonal projection to a 1-dimensional linear subspace LR3.

  1. List the eigenvalues of T.
  2. Write the characteristic polynomial pT(x) for T.
  3. Is T diagonalizable? Justify your answer.

Let L be the line L parameterized by L(t)=(2t,3t,t) for tR, and let T:R3R3 be the linear transformation that is orthogonal projection onto L.

  1. Describe ker(T) and im(T), either implicitly (using equations in x,y,z) or parametrically.
  2. List the eigenvalues of T and their geometric multiplicities.
  3. Find a basis for each eigenspace of T.
  4. Let A be the matrix for T with respect to the standard basis. Find a diagonal matrix B and an invertible matrix S such that B=S1AS. (You do not have to compute A.)

Let T:R4R4 be orthogonal projection to the 2-dimensional plane P spanned by the vectors v=(2,0,1,0) and w=(1,0,2,0).

  1. Find (with proof) all eigenvalues and eigenvectors, along with their geometric and algebraic multiplicities.
  2. Find the matrix representing T with respect to the standard basis. Is this matrix diagonalizable? Why or why not?

Let a,bR and T:R3R3 be the linear transformation that is orthogonal projection onto the plane z=ax+by (with respect to the usual Euclidean inner-product on R3).

  1. Find the eigenvalues of T and bases for the corresponding eigenspaces.
  2. Is T diagonalizable? Justify.
  3. What is the characteristic polynomial of T?

Let T:R3R3 be the orthogonal projection onto the plane z=x+y, with respect to the standard Euclidean inner product.

  1. Write the matrix representation of T with respect to the standard basis.
  2. Is T diagonalizable? Justify your answer.

Let T:R3R3 be the linear transformation that expands radially by a factor of three around the line parameterized by L(t)=[221]t, leaving the line itself fixed (viewed as a subspace).

  1. Find an eigenbasis for T and provide the matrix representation of T with respect to that basis.
  2. Provide the matrix representation of T with respect to the standard basis.

Let a,bR and T:R3R3 be the linear transformation which is reflection across the plane z=ax+by.

  1. Find the eigenvalues of T and for each find a basis for the corresponding eigenspace.
  2. Is T diagonalizable? Justify.
  3. What is the characteristic polynomial of T?
  4. What is the minimal polynomial of T?

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