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.

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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