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Linear Algebra 1st Edition Jim Hefferon - Solutions

For each real number Î¸ let tÎ¸: R2 †’ R2 be represented with respect to the standard bases by this matrix.Show that tÎ¸1+Î¸2 = tÎ¸1 ˆ™ tÎ¸2. Show also that tÎ¸-1 = t-Î¸.
Do the calculations for the proof of Corollary 4.11.Corollary 4.11The inverse for a 2 Ã— 2 matrix exists and equalsif and only if ad - bc ‰ 0.
Show that this matrixhas infinitely many right inverses. Show also that it has no left inverse.
In the review of inverses example, starting this subsection, how many left inverses has
If a matrix has infinitely many right-inverses, can it have infinitely many left-inverses? Must it have?
Assume that g: V → W is linear. One of these is true, the other is false. Which is which? (a) If f: W → V is a left inverse of g then f must be linear. (b) If f: W → V is a right inverse of g then f must be linear.
Show that if T is square and if T4 is the zero matrix then (I - T)-1 = I + T + T2 + T3. Generalize.
Let D be diagonal. Describe D2, D3, . . . , etc. Describe D-1, D-2, . . . , etc. Define D0 appropriately.
Prove that any matrix row-equivalent to an invertible matrix is also invertible.
The first question below appeared as Exercise 30. (a) Show that the rank of the product of two matrices is less than or equal to the minimum of the rank of each. (b) Show that if T and S are square then TS = I if and only if ST = I.
Perform the indicated operations, if defined.(a)(b) (c) (d) (e)
The first two parts of this question appeared as Exercise 27. (a) Show that (GH)T = HTGT. (b) A square matrix is symmetric if each i, j entry equals the j, i entry (that is, if the matrix equals its transpose). Show that the matrices HHT and HTH are symmetric. (c) Show that the inverse of the
The items starting this question appeared as Exercise 32. (a) Prove that the composition of the projections πx, πy: R3 → R3 is the zero map despite that neither is the zero map. (b) Prove that the composition of the derivatives d2/dx2, d3/dx3: P4 → P4 is the zero map despite that neither map
Is the relation 'is a two-sided inverse of' transitive? Reflexive? Symmetric?
If the sum of the elements of each row of a square matrix is k, then the sum of the elements in each row of the inverse matrix is 1/k.
Prove Theorem 1.4. Let h, g: V → W be linear maps represented with respect to bases B,D by the matrices H and G and let r be a scalar. Then with respect to B,D the map r ∙ h: V → W is represented by rH and the map h + g: V → W is represented by H + G. (a) Prove that matrix addition
For the bases in Exercise 8, find the change of basis matrix in the other direction, from D to B.Find the change of basis matrix for B, D Š† R2.(a) B = Îµ2, D = (2, 1)(b)(c) (d)
Decide if each changes bases on R2. To what basis is Îµ2 changed?(a)(b) (c) (d)
For each space find the matrix changing a vector representation with respect to B to one with respect to D.(a)(b) (c) V = P2, B = (x2, x2 + x, x2 + x + 1), D = (2, - x, x2)
Find bases such that this matrix represents the identity map with respect to those bases.
Consider the vector space of real-valued functions with basis (sin(x), cos(x)). Show that (2 sin(x) + cos(x), 3 cos(x)) is also a basis for this space. Find the change of basis matrix in each direction.
Where does this matrixsend the standard basis for R2? Any other bases?
Use Theorem 2.6 to show that a square matrix is nonsingular if and only if it is equivalent to an identity matrix.
Find the canonical representative of the matrix equivalence class of each matrix.(a)(b)
Finish the proof of Lemma 1.5. A matrix changes bases if and only if it is nonsingular.
Let H be an n × n nonsingular matrix. What basis of Rn does H change to the standard basis?
(a) In P3 with basis B = (1 + x, 1 - x, x2 + x3, x2 - x3) we have this representation.Find a basis D giving this different representation for the same polynomial. (b) State and prove that we can change any nonzero vector representation to any other.
Show that matrix equivalence is an equivalence relation.
Show that the columns of an n × n change of basis matrix form a basis for Rn. Do all bases appear in that way: can the vectors from any Rn basis make the columns of a change of basis matrix?
Find a matrix having this effect.That is, find a M that left-multiplies the starting vector to yield the ending vector. Is there a matrix having these two effects? (a) (b) Give a necessary and sufficient condition for there to be a matrix such that 1 †’ 1 and 2 †’ 2.
How many matrix equivalence classes are there?
Are matrix equivalence classes closed under scalar multiplication? Addition?
Let t : Rn †’ Rn represented by T with respect to Îµn, Îµn.(a) Find RepB,B(t) in this specific case.(b) Describe RepB,B(t) in the general case where B = (1, ..., n).
(a) Let V have bases B1 and B2 and suppose that W has the basis D. Where h: V → W, find the formula that computes RepB2,D(h) from RepB1,D(h). (b) Repeat the prior question with one basis for V and two bases for W.
(a) If two matrices are matrix equivalent and invertible, must their inverses be matrix equivalent? (b) If two matrices have matrix equivalent inverses, must the two be matrix equivalent? (c) If two matrices are square and matrix equivalent, must their squares be matrix equivalent? (d) If two
Square matrices are similar if they represent the same transformation, but each with respect to the same ending as starting basis. That is, RepB1,B1 (t) is similar to RepB2,B2 (t). (a) Give a definition of matrix similarity like that of Definition 2.3. (b) Prove that similar matrices are matrix
Suppose that, with respect tothe transformation t : R2 †’ R2 is represented by this matrix. Use change of basis matrices to represent t with respect to each pair. (a) (b)
Consider the spaces V = P2 and W = M2Ã—2, with these bases.We will find P and Q to convert the representation of a map with respect to B,D to one with respect to , (a) Draw the appropriate arrow diagram. (b) Compute P and Q.
Find the P and Q to express H via PHQ as a block partial identity matrix.
Why does Theorem 2.6 not show that every matrix is diagonalizable (see Example 2.2)?Any m Ã— n matrix of rank k is matrix equivalent to the m Ã— n matrix that is all zeros except that the first k diagonal entries are ones.
Show that a zero matrix is alone in its matrix equivalence class. Are there other matrices like that?
Decide if these matrices are matrix equivalent.(a)(b) (c)
In R2, wherefind the change of basis matrices from D to Îµ2 and from Îµ2 to D. Multiply the two.
Find the change of basis matrix for B,D Š† R2.(a) B = Îµ2, D = (2, 1)(b)(c) (d)
Find the change of basis matrix for each B,D ⊆ P2. (a) B = (1, x, x2),D = (x2, 1, x) (b) B = (1, x, x2),D = (1, 1 + x, 1 + x + x2) (c) B = (2, 2x, x2),D = (1 + x2, 1 - x2, x + x2)
Example 1.4 suggests that projection breaks into two parts and that are non-interacting. Recall that the two are orthogonal. Show that any two nonzero orthogonal vectors make up a linearly independent set.
(a) What is the orthogonal projection of into a line if is a member of that line?(b) Show that if is not a member of the line then the set is linearly independent.
Definition 1.1 requires that ~s be nonzero. Why? What is the right definition of the orthogonal projection of a vector into the (degenerate) line spanned by the zero vector?
Are all vectors the projection of some other vector into some line?
Show that the projection of into the line spanned by has length equal to the absolute value of the number divided by the length of the vector .
Find the formula for the distance from a point to a line.
Find the scalar c such that the point (cs1; cs2) is a minimum distance from the point (v1, v2) by using calculus (i.e., consider the distance function, set the first derivative equal to zero, and solve). Generalize to Rn.
Prove that the orthogonal projection of a vector into a line is shorter than the vector.
Find a nonzero vector in R3 that is orthogonal to both of these.
Consider the function mapping the plane to itself that takes a vector to its projection into the line y = x. These two each show that the map is linear, the first one in a way that is coordinate-bound (that is, it fixes a basis and then computes) and the second in a way that is more conceptual.(a)
For be the projection of into the line spanned by be the projection of into the line spanned by be the projection of into the line spanned by etc., back and forth between the spans of That is, is the projection of into the span of if i + 1 is even, and into the span of if i + 1 is odd.
Perform the Gram-Schmidt process on each of these bases for R2.(a)(b) (c) Then turn those orthogonal bases into orthonormal bases.
Show that the columns of an n×n matrix form an orthonormal set if and only if the inverse of the matrix is its transpose. Produce such a matrix.
Does the proof of Theorem 2.2 fail to consider the possibility that the set of vectors is empty (i.e., that k = 0)?
(a) Prove that the matrix RepK,B(id) changing bases in the direction opposite to that of the theorem has an upper triangular shape-all of its entries below the main diagonal are zeros.(b) Prove that the inverse of an upper triangular matrix is also upper triangular (if the matrix is invertible,
Complete the induction argument in the proof of Theorem 2.7.
Perform the Gram-Schmidt process on each of these bases for R3.(a)(b) Then turn those orthogonal bases into orthonormal bases.
Find an orthonormal basis for this subspace of R3: the plane x - y + z = 0.
Find an orthonormal basis for this subspace of R4.
Show that any linearly independent subset of Rn can be orthogonalized without changing its span.
What happens if we try to apply the Gram-Schmidt process to a finite set that is not a basis?
(a) Prove that for any ~v in the space, the vector
One advantage of orthogonal bases is that they simplify finding the representation of a vector with respect to that basis.(a) For this vector and this non-orthogonal basis for R2(b) With this orthogonal basis for R2
Bessel's Inequality. Consider these orthonormal sets
A square matrix is symmetric if each i; j entry equals the j; i entry (i.e., if the matrix equals its transpose). Show that the projection matrix A(ATA)-1AT is symmetric.
Project the vectors into M along N.(a)(b) (c)
Find M”´.(a)(b) (c) (d) (e) (f) (g)
This subsection shows how to project orthogonally in two ways, the method of Example 3.2 and 3.3, and the method of Theorem 3.8. To compare them, consider the plane P specified by 3x + 2y - z = 0 in R3.(a) Find a basis for P.(b) Find P”´ and a basis for P”´.(c) Represent this
We have three ways to find the orthogonal projection of a vector into a line, the Definition 1.1 way from the first subsection of this section, the Example 3.2 and 3.3 way of representing the vector with respect to a basis for the space and then keeping the M part, and the way of Theorem 3.8. For
Check that the operation of Definition 3.1 is well-defined. That is, in Example 3.2 and 3.3, doesn't the answer depend on the choice of bases?
What is the orthogonal projection into the trivial subspace?
What is the projection of v into M along if ?
Show that if is a subspace with orthonormal basis then the orthogonal projection of into M is this.
Prove that the map p: V ! V is the projection into M along N if and only if the map id-p is the projection into N along M. (Recall the definition of the difference of two maps: (id-p)
Show that if a vector is perpendicular to every vector in a set then it is perpendicular to every vector in the span of that set.
Suppose that are such that for all complements the projections of into M along N are equal. Must (If so, what if we relax the condition to: all orthogonal projections of the two are equal?)
Let M,N be subspaces of Rn. The perp operator acts on subspaces, we can ask how it interacts with other such operations.(a) Show that two perps cancel: (M┴)┴ = M.(b) Prove that M N implies that N┴ ⊂ M┴.(c) Show that (M+ N)┴ = M┴ ∩ N?.
The material in this subsection allows us to express a geometric relationship that we have not yet seen between the range space and the null space of a linear map.(a) Represent f : R3 †’ R given bywith respect to the standard bases and show thatis a member of the perp of the null space. Prove
(a) Show that orthogonal projection into a line has that property.(b) Show that projection along a subspace has that property.
Project the first vector orthogonally into the line spanned by the second vector.(a)(b) (c) (d)
Project the vector orthogonally into the line.(a)(b)
Although pictures guided our development of Definition 1.1, we are not restricted to spaces that we can draw. In R4 project this vector into this line.
(a)
Use least-squares to judge if the coin in this experiment is fair.
What combination of dilations, flips, skews, and projections produces a rotation counterclockwise by 2π/3 radians?
What combination of dilations, flips, skews, and projections produces the map h: R3 †’ R3 represented with respect to the standard bases by this matrix?
Show that for any permutation (that is, reordering) p of the numbers 1, . . . , n, the mapcan be done with a composition of maps, each of which only swaps a single pair of coordinates.
Show that linear maps preserve the linear structures of a space.(a) Show that for any linear map from Rn to Rm, the image of any line is a line. The image may be a degenerate line, that is, a single point.(b) Show that the image of any linear surface is a linear surface. This generalizes the result
Use a picture like the one that appears in the discussion of the Chain Rule to answer: if a function f : R → R has an inverse, what's the relationship between how the function-locally, approximately-dilates space, and how its inverse dilates space (assuming, of course, that it has an inverse)?
Let M be a 3×3 magic square with magic number s. (a) Prove that the sum of M's entries is 3s. (b) Prove that s = 3 ∙ m2,2. (c) Prove that m2,2 is the average of the entries in its row, its column, and in each diagonal. (d) Prove that m2,2 is the median of M's entries.
Solve the system a+b = s, c+d = s, a+c = s, b+d = s, a+d = s, and b+c = s.
Let the trace function be Tr(M) = m1,1 + ∙∙ ∙ +mn,n. Define also the sum down the other diagonal Tr*(M) = m1,n + ∙∙ ∙+ mn,1. (a) Show that the two functions Tr,Tr* : Mn×n → R are linear. (b) Show that the function θ: Mn×n → R2 given by θ(M) = (Tr(M),Tr*(m)) is linear. (c)
A square matrix is semimagic if the rows and columns add to the same value, that is, if we drop the condition on the diagonals. (a) Show that the set of semimagic squares Hn is a subspace of Mn×n. (b) Show that the set Hn,0 of n×n semimagic squares with magic number 0 is also a subspace of Mn×n.
We consider throws of a die, and say the system is in state si if the largest number yet appearing on the die was i. (a) Give the transition matrix. (b) Start the system in state s1, and run it for five throws. What is the vector at the end?
For the men's mile record, rather than give each of the many records and its exact date, we've "smoothed" the data somewhat by taking a periodic sample. Do the longer calculation and compare the conclusions.

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