- Show the map T : ℝ
^{n}→ ℝ^{m}defined by T= Ax where A is an m × n matrix and x is an m × 1 column vector is a linear transformation. - Find the matrix for the linear transformation which rotates every vector in ℝ
^{2}through an angle of π∕3. - Find the matrix for the linear transformation which rotates every vector in ℝ
^{2}through an angle of π∕4. - Find the matrix for the linear transformation which rotates every vector in ℝ
^{2}through an angle of −π∕3. - Find the matrix for the linear transformation which rotates every vector in ℝ
^{2}through an angle of 2π∕3. - Find the matrix for the linear transformation which rotates every vector in ℝ
^{2}through an angle of π∕12. Hint: Note that π∕12 = π∕3 − π∕4. - Find the matrix for the linear transformation which rotates every vector in ℝ
^{2}through an angle of 2π∕3 and then reflects across the x axis. - Find the matrix for the linear transformation which rotates every vector in ℝ
^{2}through an angle of π∕3 and then reflects across the x axis. - Find the matrix for the linear transformation which rotates every vector in ℝ
^{2}through an angle of π∕4 and then reflects across the x axis. - Find the matrix for the linear transformation which rotates every vector in ℝ
^{2}through an angle of π∕6 and then reflects across the x axis followed by a reflection across the y axis. - Find the matrix for the linear transformation which reflects every vector in ℝ
^{2}across the x axis and then rotates every vector through an angle of π∕4. - Find the matrix for the linear transformation which rotates every vector in ℝ
^{2}through an angle of π∕4 and next reflects every vector across the x axis. Compare with the above problem. - Find the matrix for the linear transformation which reflects every vector in ℝ
^{2}across the x axis and then rotates every vector through an angle of π∕6. - Find the matrix for the linear transformation which reflects every vector in ℝ
^{2}across the y axis and then rotates every vector through an angle of π∕6. - Find the matrix for the linear transformation which rotates every vector in ℝ
^{2}through an angle of 5π∕12. Hint: Note that 5π∕12 = 2π∕3 − π∕4. - Find the matrix for proj
_{u}where u =^{T}. - Find the matrix for proj
_{u}where u =^{T}. - Find the matrix for proj
_{u}where u =^{T}. - Give an example of a 2 × 2 matrix A which has all its entries nonzero and satisfies
A
^{2}= A. A matrix which satisfies A^{2}= A is called idempotent. - Let A be an m × n matrix and let B be an n × m matrix where n < m. Show that AB cannot have an inverse.
- Find kerfor
Recall ker

is just the set of solutions to Ax = 0. - If A is a linear transformation, and Ax
_{p}= b, show that the general solution to the equation Ax = b is of the form x_{p}+ y where y ∈ ker. By this I mean to show that whenever Az = b there exists y ∈ kersuch that x_{p}+ y = z. For the definition of kersee Problem 21. - Using Problem 21, find the general solution to the following linear system.
- Using Problem 21, find the general solution to the following linear system.
- Show that the function T
_{u}defined by T_{u}≡ v − proj_{u}is also a linear transformation. - If u =
^{T}, as in Example 2.4.5 and T_{u}is given in the above problem, find the matrix A_{u}which satisfies A_{u}x = T_{u}. - Let a be a fixed vector. The function T
_{a}defined by T_{a}v = a + v has the effect of translating all vectors by adding a. Show this is not a linear transformation. Explain why it is not possible to realize T_{a}in ℝ^{3}by multiplying by a 3 × 3 matrix. - In spite of Problem 27 we can represent both translations and rotations by matrix
multiplication at the expense of using higher dimensions. This is done by the homogeneous
coordinates. I will illustrate in ℝ
^{3}where most interest in this is found. For each vector v =^{T}, consider the vector in ℝ^{4}^{T}. What happens when you doDescribe how to consider both rotations and translations all at once by forming appropriate 4 × 4 matrices.

- You want to add to every point in ℝ
^{3}and then rotate about the x axis clockwise through the angle of 30^{∘}. Find what happens to the point. - You are given a linear transformation T : F
^{n}→ F^{m}and you know thatwhere

^{−1}exists. Show that the matrix A of T with respect to the usual basis vectors (Ax = Tx) must be of the form - You have a linear transformation T and
- You have a linear transformation T and
- You have a linear transformation T and
- You have a linear transformation T and
- You have a linear transformation T and
- Suppose V is a subspace of F
^{n}and T : V → F^{p}is a nonzero linear transformation. Show that there exists a basis for Im≡ T(V )and that in this situation,

is linearly independent.

- ↑In the situation of Problem 36 where V is a subspace of F
^{n}, show that there existsa basis for ker. (Recall Theorem 2.6.12. Since keris a subspace, it has a basis.) Now for an arbitrary Tv ∈ T, explain whyand why this implies

Then explain why V = span

. - ↑In the situation of the above problem, show is a basis for V and therefore, dim= dim+ dim.
- ↑Let A be a linear transformation from V to W and let B be a linear transformation from
W to U where V,W,U are all subspaces of some F
^{p}. Explain why - ↑Let be a basis of kerand letbe a basis of A. Let z ∈ ker. Explain why
and why there exist scalars a

_{i}such thatand why it follows z −

∈ span. Now explain whyand so

This important inequality is due to Sylvester. Show that equality holds if and only if A(kerBA) = ker(B).

- Generalize the result of the previous problem to any finite product of linear mappings.
- If W ⊆ V for W,V two subspaces of F
^{n}and if dim= dim, show W = V . - Let V be a subspace of F
^{n}and let V_{1},,V_{m}be subspaces, each contained in V . Then(2.37) if every v ∈ V can be written in a unique way in the form

where each v

_{i}∈ V_{i}. This is called a direct sum. If this uniqueness condition does not hold, then one writesand this symbol means all vectors of the form

Show 2.37 is equivalent to saying that if

then each v

_{j}= 0. Next show that in the situation of 2.37, if β_{i}=is a basis for V_{i}, thenis a basis for V . - ↑Suppose you have finitely many linear mappings L
_{1},L_{2},,L_{m}which map V to V where V is a subspace of F^{n}and suppose they commute. That is, L_{i}L_{j}= L_{j}L_{i}for all i,j. Also suppose L_{k}is one to one on kerwhenever j≠k. Letting P denote the product of these linear transformations, P = L_{1}L_{2}L_{m}, first showNext show L

_{j}: ker→ ker. Then showUsing Sylvester’s theorem, and the result of Problem 42, show

Hint: By Sylvester’s theorem and the above problem,

- Let ℳdenote the set of all n × n matrices having entries in F. With the usual operations of matrix addition and scalar multiplications, explain why ℳcan be considered as F
^{n2 }. Give a basis for ℳ. If A ∈ℳ, explain why there exists a monic (leading coefficient equals 1) polynomial of the formsuch that

The minimal polynomial of A is the polynomial like the above, for which p

= 0 which has smallest degree. I will discuss the uniqueness of this polynomial later. Hint: Consider the matrices I,A,A^{2},,A^{n2 }. There are n^{2}+ 1 of these matrices. Can they be linearly independent? Now consider all polynomials and pick one of smallest degree and then divide by the leading coefficient. - ↑Suppose the field of scalars is ℂ and A is an n × n matrix. From the preceding problem,
and the fundamental theorem of algebra, this minimal polynomial factors
where r

_{j}is the algebraic multiplicity of λ_{j}, and the λ_{j}are distinct. Thusand so, letting P =

^{r1}^{r2}^{rk}and L_{j}=^{rj}apply the result of Problem 44 to verify thatand that A : ker

→ ker. In this context, keris called the generalized eigenspace for λ_{j}. You need to verify the conditions of the result of this problem hold. - In the context of Problem 46, show there exists a nonzero vector x such that
This is called an eigenvector and the λ

_{j}is called an eigenvalue. Hint:There must exist a vector y such thatWhy? Now what happens if you do

to z? - Suppose Qis an orthogonal matrix. This means Qis a real n × n matrix which satisfies
Suppose also the entries of Q

are differentiable. Show^{′}= −Q^{T}Q^{′}Q^{T}. - Remember the Coriolis force was 2Ω × v
_{B}where Ω was a particular vector which came from the matrix Qas described above. Show thatThere will be no Coriolis force exactly when Ω = 0 which corresponds to Q

^{′}= 0 . When will Q^{′}= 0? - An illustration used in many beginning physics books is that of firing a rifle horizontally and
dropping an identical bullet from the same height above the perfectly flat ground followed by
an assertion that the two bullets will hit the ground at exactly the same time. Is this true on
the rotating earth assuming the experiment takes place over a large perfectly flat field so the
curvature of the earth is not an issue? Explain. What other irregularities will occur? Recall
the Coriolis acceleration is 2ωwhere k points away from the center of the earth, j points East, and i points South.

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