- Let be a basis for F
^{n}and define a mapping T : F^{n}→ spanas follows.Explain why this is a linear transformation.

- In the above problem, suppose v
_{k}= u_{k}. Show that Tv = v if v ∈ V ≡ span. Now show that T= Tand that≤. - Find the minimum polynomials for the following matrices and use to obtain the eigenvalues of the
matrix. The set of all eigenvectors associated with an eigenvalue λ is called the eigenspace. Determine
the eigenspaces for each of these matrices.
- Suppose you have pis the minimum polynomial for a square n × n matrix A. Show that this matrix is invertible if and only if the constant term of the minimum polynomial is non zero. In this case, give a formula for A
^{−1}in terms of powers of A. SayThus you need explain why a

_{0}≠0 if A^{−1}exists and then find a formula for A^{−1}when this is the case. - Find least squares solutions to the following systems of equations.
- =
- =
- =

- Here are some matrices. Label according to whether they are symmetric, skew symmetric, or
orthogonal. If the matrix is orthogonal, determine whether it is proper or improper.
(a)

(b)(c) - Show that every real matrix may be written as the sum of a skew symmetric and a symmetric matrix.
Hint: If A is an n × n matrix, show that B ≡is skew symmetric.
- Let x be a vector in ℝ
^{n}and consider the matrix I −. Show this matrix is both symmetric and orthogonal. - For U an orthogonal matrix, explain why =for any vector x. Next explain why if U is an n × n matrix with the property that=for all vectors, x, then U must be orthogonal. Thus the orthogonal matrices are exactly those which preserve distance. This was done in general in the chapter for unitary matrices. Do it here for the special case that the matrix is orthogonal. It will be simpler.
- A quadratic form in three variables is an expression of the form a
_{1}x^{2}+ a_{2}y^{2}+ a_{3}z^{2}+ a_{4}xy + a_{5}xz + a_{6}yz. Show that every such quadratic form may be written aswhere A is a symmetric matrix.

- Given a quadratic form in three variables, x,y, and z, show there exists an orthogonal matrix U and
variables x
^{′},y^{′},z^{′}such that^{T}= U^{T}with the property that in terms of the new variables, the quadratic form iswhere the numbers, λ

_{1},λ_{2}, and λ_{3}are the eigenvalues of the matrix A in Problem 10. - If A is a symmetric invertible matrix, is it always the case that A
^{−1}must be symmetric also? How about A^{k}for k a positive integer? Explain. - If A,B are symmetric matrices, does it follow that AB is also symmetric?
- Suppose A,B are symmetric and AB = BA. Does it follow that AB is symmetric?
- Here are some matrices. What can you say about the eigenvalues of these matrices just by looking at
them?
(a)

(b)(c)(d) - Find the eigenvalues and eigenvectors of the matrix . Here b,c are real numbers.
- Find the eigenvalues and eigenvectors of the matrix . Here a,b,c are real numbers.
- Find the eigenvalues and an orthonormal basis of eigenvectors for A.
Hint: Two eigenvalues are 12 and 18.

- Find the eigenvalues and an orthonormal basis of eigenvectors for A.
Hint: One eigenvalue is 3.

- Show that if A is a real symmetric matrix and λ and μ are two different eigenvalues, then if x is an
eigenvector for λ and y is an eigenvector for μ, then x ⋅ y = 0. Also all eigenvalues are real. Supply
reasons for each step in the following argument. First
and so λ = λ. This shows that all eigenvalues are real. It follows all the eigenvectors are real. Why? Now let x,y,μ and λ be given as above.

and so

Since λ≠μ, it follows x ⋅ y = 0.

- Suppose U is an orthogonal n × n matrix. Explain why rank = n.
- Show that if A is an Hermitian matrix and λ and μ are two different eigenvalues, then if x is an
eigenvector for λ and y is an eigenvector for μ, then = 0 . Also all eigenvalues are real. Supply reasons for each step in the following argument. First
and so λ = λ. This shows that all eigenvalues are real. Now let x,y,μ and λ be given as above.

and so

= 0 . Since λ≠μ, it follows= 0 . - Show that the eigenvalues and eigenvectors of a real matrix occur in conjugate pairs.
- If a real matrix A has all real eigenvalues, does it follow that A must be symmetric. If so, explain why and if not, give an example to the contrary.
- Suppose A is a 3 × 3 symmetric matrix and you have found two eigenvectors which form an orthonormal set. Explain why their cross product is also an eigenvector.
- Determine which of the following sets of vectors are orthonormal sets. Justify your answer.
(a)

(b)(c) - Show that if is an orthonormal set of vectors in F
^{n}, then it is a basis. Hint: It was shown earlier that this is a linearly independent set. - Fill in the missing entries to make the matrix orthogonal.
- Fill in the missing entries to make the matrix orthogonal.
- Fill in the missing entries to make the matrix orthogonal.
- Find the eigenvalues and an orthonormal basis of eigenvectors for A. Diagonalize A by finding an
orthogonal matrix U and a diagonal matrix D such that U
^{T}AU = D.Hint: One eigenvalue is -2.

- Find the eigenvalues and an orthonormal basis of eigenvectors for A. Diagonalize A by finding an
orthogonal matrix U and a diagonal matrix D such that U
^{T}AU = D.Hint: Two eigenvalues are 18 and 24.

- Find the eigenvalues and an orthonormal basis of eigenvectors for A. Diagonalize A by finding an
orthogonal matrix U and a diagonal matrix D such that U
^{T}AU = D.Hint: Two eigenvalues are 12 and 18.

- Find the eigenvalues and an orthonormal basis of eigenvectors for A. Diagonalize A by finding an
orthogonal matrix U and a diagonal matrix D such that U
^{T}AU = D.Hint: The eigenvalues are −3,−2,1.

- Find the eigenvalues and an orthonormal basis of eigenvectors for A. Diagonalize A by finding an
orthogonal matrix U and a diagonal matrix D such that U
^{T}AU = D. - Find the eigenvalues and an orthonormal basis of eigenvectors for A. Diagonalize A by finding an
orthogonal matrix U and a diagonal matrix D such that U
^{T}AU = D. - Find the eigenvalues and an orthonormal basis of eigenvectors for A. Diagonalize A by finding an
orthogonal matrix U and a diagonal matrix D such that U
^{T}AU = D.Hint: The eigenvalues are 0,2,2 where 2 is listed twice because it is a root of multiplicity 2.

- Find the eigenvalues and an orthonormal basis of eigenvectors for A. Diagonalize A by finding an
orthogonal matrix U and a diagonal matrix D such that U
^{T}AU = D.Hint: The eigenvalues are 2,1,0.

- Find the eigenvalues and an orthonormal basis of eigenvectors for the matrix
Hint: The eigenvalues are 1,2,−2.

- Find the eigenvalues and an orthonormal basis of eigenvectors for the matrix
Hint: The eigenvalues are −1,2,−1 where −1 is listed twice because it has multiplicity 2 as a zero of the characteristic equation.

- Explain why a real matrix A is symmetric if and only if there exists an orthogonal matrix U such
that A = U
^{T}DU for D a diagonal matrix. - You are doing experiments and have obtained the ordered pairs,
Find m and b such that y = mx + b approximates these four points as well as possible. Now do the same thing for y = ax

^{2}+ bx + c, finding a,b, and c to give the best approximation. - Suppose you have several ordered triples, . Describe how to find a polynomial,
for example giving the best fit to the given ordered triples. Is there any reason you have to use a polynomial? Would similar approaches work for other combinations of functions just as well?

- Find an orthonormal basis for the spans of the following sets of vectors.
- ,,.
- ,,
- ,,

- Using the Gram Schmidt process or the QR factorization, find an orthonormal basis for the span of
the vectors, ,, and.
- Using the Gram Schmidt process or the QR factorization, find an orthonormal basis for the span of
the vectors, ,, and.
- The set, V ≡is a subspace of ℝ
^{3}. Find an orthonormal basis for this subspace. - The two level surfaces, 2x + 3y − z + w = 0 and 3x − y + z + 2w = 0 intersect in a subspace of ℝ
^{4}, find a basis for this subspace. Next find an orthonormal basis for this subspace. - Let A,B be a m × n matrices. Define an inner product on the set of m × n matrices
by
Show this is an inner product satisfying all the inner product axioms. Recall for M an n×n matrix, trace

≡∑_{i=1}^{n}M_{ii}. The resulting norm,_{F}is called the Frobenius norm and it can be used to measure the distance between two matrices. - Let A be an m×n matrix. Show
_{F}^{2}≡_{F}= ∑_{j}σ_{j}^{2}where the σ_{j}are the singular values of A. - The trace of an n×n matrix M is defined as ∑
_{i}M_{ii}. In other words it is the sum of the entries on the main diagonal. If A,B are n × n matrices, show trace= trace. Now explain why if A = S^{−1}BS it follows trace= trace. Hint: For the first part, write these in terms of components of the matrices and it just falls out. - Using Problem 51 and Schur’s theorem, show that the trace of an n×n matrix equals the sum of the eigenvalues.
- If A is a general n×n matrix having possibly repeated eigenvalues, show there is a sequence of n×n matrices having distinct eigenvalues which has the property that the ij
^{th}entry of A_{k}converges to the ij^{th}entry of A for all ij. Hint: Use Schur’s theorem.

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