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M 340L (Fall 2007, 60155)
Due: Friday, October 12, 2007 1
- Let f : Rn^ −→ Rm^ be linear map. Let
ker(f ) := { x ∈ Rn^ | f (x) = 0 }, and
image(f ) := { f (x) ∈ Rm^ | x ∈ Rn^ } = { y ∈ Rm^ | y = f (x) for some x ∈ Rn^ }.
ker(f ) is called the kernel of f and image(f ) is called the image (or range) of f.
a) Prove that the kernel of a linear map is a subspace of the domain of the linear map. Hint: Let f : Rn^ −→ Rm^ be linear.
- ker(f ) must be non-empty since it must contain.
- Closedness of ker(f ) under addition: Suppose x 1 , x 2 ∈ ker(f ), i.e. f (x 1 ) = 0 and f (x 2 ) = 0. Show that x 1 + x 2 ∈ ker(f ), i.e. f (x 1 + x 2 ) = 0.
- Closedness of ker(f ) under scalar multiplication: Suppose α ∈ R and x ∈ ker(f ), i.e. f (x) = 0. Show that α x ∈ ker(f ), i.e. f (α x) = 0.
b) Prove that the image of a linear map is a subspace of the codomain of the linear map. Hint: Let f : Rn^ −→ Rm^ be linear.
- image(f ) must be non-empty since it must contain.
- Closedness of image(f ) under addition AND scalar multiplication: Suppose α ∈ R and y 1 , y 2 ∈ image(f ), i.e. there exist x 1 , x 2 ∈ Rn^ such that f (x 1 ) = y 1 and f (x 2 ) = y 2. Show that y 1 + α y 2 ∈ image(f ), i.e. there exists x 3 ∈ Rn^ such that f (x 3 ) = y 1 + α y 2. Use the linearity of f to find such an x 3 from α, x 1 and x 2.
c) Find the kernel and image of proj : R^2 −→ R^2 : x 7 −→
» (^1 ) 0 0
Hint: Watch how theory can free you from computing: For the kernel, note that ker(proj) is a subspace of R^2 ; hence, it has a dimension, which can only be (why?) either 0 , 1 , or 2. Which one?
- Can dim ker(proj) be two? Note that if a linear map R^2 −→ Rwhatever^ has a 2 -dimensional kernel, then it maps all of R^2 to one single vector (what is it?). Again, does proj have a 2 -dimensional kernel?
- Can dim ker(proj) be zero? Note that dim ker(proj) = 0 ⇐⇒ ker(proj) = { 0 } (why?). Is the latter statement true? In other words, does proj map only 0 , and no other vector, to 0?
So by now, you should know what the dimension of ker(proj) must be. Can you find a basis for it? For the image, note that
n„ (^1) 0
« ,
„ (^0) 1
«o is a basis for domain(proj) = R^2. This implies (why?) that image(proj) = { proj(x) ∈
R^2 | x ∈ R^2 } =
n proj
x
„ (^1) 0
«
„ (^0) 1
∈ R^2
˛ x, y^ ∈^ R
o = · · · = span
n proj
„ (^1) 0
« , proj
„ (^0) 1
«o = span{? ,? }. You should be able to complete your solution from here.
d) Find the kernel and image of f : R^2 −→ R^2 : x 7 −→
» 3 1 1 3
(For parts (a) and (b), use only the definitions of kernel, image, linearity, and subspace. In particular, do NOT use the fact a linear map from Rn^ to Rm^ must be left-multiplication by some matrix in Rm×n. The reason for this restriction is that the “subspace-hood” of the kernel and image of a linear map is a consequence of linearity alone, and remains true for linear maps with more general domains and codomains than Rn^ and Rm, as we will shortly see.)
- Read §2.1 of the course text.
a) Never forget this: Cancellation law does NOT hold for matrix multiplication. This exercise illustrates the reason. Recall the fact that for x, y ∈ R, the equality xy = 0 implies either x = 0 or y = 0. Consequently, suppose a, b, c ∈ R and a 6 = 0. Then, ab = ac ⇐⇒ a(b − c) = 0 =⇒ b − c = 0, i.e. b = c. (hence, you have cancelled the nonzero a from ab = ac.) This is the basis for the cancellation law in the arithmetic of real numbers. The analogous statement for matrix multiplication is false.
M 340L (Fall 2007, 60155)
Due: Friday, October 12, 2007 2
Give an example of a pair of nonzero matrices A, B ∈ R^2 ×^2 such that AB is the 2 × 2 zero matrix.
(Hint: Simply take A to be
» 1 0 0 0
- . Move the entry 1 to a different spot in order to obtain a suitable B ∈ R^2 ×^2 .)
b) Let A =
[ 2 3 4 5
]
and B =
[ 6 7 8 9
]
. Compute (AB)T^ and BT^ AT^. (Note that they are equal.)
c) Let A ∈ R^2 ×^3 and B ∈ R^3 ×^4. Then, A B ∈ R^2 ×^4 is well-defined, and hence so is (A B)T^ ∈ R^4 ×^2. On the other hand, BT^ ∈ R^4 ×^3 and AT^ ∈ R^3 ×^2 ; hence BT^ AT^ ∈ R^4 ×^2 has the same “shape” as (A B)T^.
Can you guess how the two matrices (A B)T^ , BT^ AT^ ∈ R^4 ×^2 are related?
d) Let
A =
[ a 11 a 12 a 13 a 21 a 22 a 23
]
∈ R^2 ×^3 and B =
b 11 b 12 b 13 b 14 b 21 b 22 b 23 b 24 b 31 b 32 b 33 b 34
(^) ∈ R^3 ×^4.
Compute the (3, 2)-entry of (A B)T^. (This is just the (2, 3)-entry of A B.) Compute the (3, 2)-entry of BT^ AT^. (Note that these two entries are equal.) (Convince yourself that (AB)T^ = BT^ AT^ ∈ Rk×m, for any A ∈ Rm×n^ and B ∈ Rn×k^ .)
- Let
A =
5 ,^ and^ B^ =
a) Let
E 1 :=
(^75) , E 2 :=
(^75) , E 3 :=
(^75).
What row operation(s) does left-multiplication by E 1 perform on 4 × n matrices? By E 2? By E 3? (Remark: The matrices E 1 , E 2 , E 3 are NOT “elementary” (see p.122); they perform more than one elementary row operation simultaneously. But each of them is a product of elementary matrices.)
b) Compute M := E 3 E 2 E 1. Verify that E 3 E 2 E 1 A = B. Explain why this proves that RREF(A) = B.
c) Is E 1 invertible? Determine E 1 − 1 , noting that left-multiplication by E 1 − 1 must undo the effect of left multiplication by E 1. Verify indeed that E 1 E 1 − 1 = I 4 = E 1 − 1 E 1.
d) Detemine E 2 − 1 and E− 3 1 the same way.
e) Compute N := E 1 − 1 E 2 − 1 E− 3 1. Verify that M as in (3b) is invertible by showing M N = I 4 = N M. (This shows indeed that (E 3 E 2 E 1 )−^1 = E 1 − 1 E 2 − 1 E 3 − 1. More generally, if A 1 , A 2 ,... , Ak ∈ Rn×n^ are all invertible, can you express (A 1 A 2 · · · Ak )−^1 in terms of A− 1 1 ,... , A− k 1? )
M 340L (Fall 2007, 60155)
Due: Friday, October 12, 2007 4
- Prove: Given any basis { v 1 ,... , vn } of Rn, every vector x ∈ Rn^ can be expressed UNIQUELY as a linear combination of v 1 ,.. ., vn.
One way to proceed:
- First of all, why can x be written as a linear combination of v 1 ,.. ., vn? (This is just a matter of definition. Look up the definition of a basis IMMEDIATELY if you need to.)
- It remains only to establish the uniqueness part of the statement. So, suppose x can be written as a linear combination of v 1 ,.. ., vn in two ways, say,
x = α 1 v 1 + · · · + αn vn = β 1 v 1 + · · · + βn vn.
We can complete the proof by showing that the “two” hypothetical ways are in fact the same way by showing that the following must hold: α 1 = β 1 ,... , αn = βn. Which property of a basis (applied to v 1 ,... , vn of course) will allow us to make this conclusion? (This requires again the definition of a basis and a little “re-arrangement of things.”)
Remark: Given any ordered basis { v 1 ,... , vn } for Rn, this result shows that
- every n-tuple (α 1 ,... , αn) of numbers in R uniquely determines a vector x ∈ Rn^ via the equation x = α 1 v 1 +· · ·+αn vn,
- and, conversely, every vector x ∈ Rn^ uniquely determines an n-tuple (α 1 ,... , αn) of real numbers via the same equation x = α 1 v 1 + · · · + αn vn.
We call the n-tuple (α 1 ,... , αn) the coordinates of x = α 1 v 1 + · · · + αn vn with respect to the ordered basis { v 1 ,... , vn }.
- Projection onto a line in the plane.
Recall that the map proj : R^2 −→ R^2 : x 7 −→
» (^1 ) 0 0
- x is the “projection” of the plane R^2 onto the “horizontal” axis. More specifically, it is the projection onto the horizontal axis in the direction perpendicular (or more formally, orthogonal) to the horizontal axis. How do we “orthogonally project” onto a 1 -dimensional subspace (i.e. line through origin) which is “not horizontal”? One way to proceed: “rotate, vertically project, rotate back.”
Let L ⊂ R^2 be the line as follows:
L := span{u} =
{ t u ∈ R^2 | t ∈ R
} , where u =
( √ 3 1
)
.
a) Let rotθ : R^2 −→ R^2 denote the linear map of counterclockwise^1 rotation by the angle θ ∈ R. Find the value of θ ∈ [0, π) such that Rθ maps the horizontal axis to L. Let θ be fixed at this value for the rest of this problem.
b) Find the matrix Rθ ∈ R^2 ×^2 whose left multiplication is the map rotθ : R^2 −→ R^2.
c) What is the image of L under rot(−θ)? (Recall: rotθ is the rotation that rotates the horizontal axis to L.)
d) Find the “standard matrix” (or “left multiplication matrix”) PL ∈ R^2 ×^2 for the linear map projL := rotθ ◦ proj ◦ rot(−θ), where proj : R^2 −→ R^2 is “orthogonal projection onto horizontal axis.” In other words, the matrix PL ∈ R^2 ×^2 is determined by projL(x) = PL x, for each x ∈ R^2.
e) Let v =
( √−^1 3
)
. Find the preimage^2 of v under rotθ.
(^1) By standard convention, a rotation by a negative angle θ means clockwise rotation by |θ| radians, where |θ| is the absolute value of θ. (^2) Let f : X −→ Y be a map from the set X to the set Y and let y ∈ Y. The preimage f − (^1) (y) of y under f is f − (^1) (y) := { x ∈ X | f (x) = y }. In other words, f −^1 (y) is simply the subset of elements of X each of which is mapped to y by f. For example, consider f : R −→ R : x 7 −→ x^2. Then, f −^1 (0) = { 0 }, f −^1 (1) = {± 1 }, f −^1 (4) = {± 2 }, and f −^1 (−1) = ∅, the empty set.
M 340L (Fall 2007, 60155)
Due: Friday, October 12, 2007 5
f) Explain why {u, v} is a basis for R^2. Describe the relation between the vectors u, v and (the columns of) the matrix Rθ.
g) Compute projL(u) and projL(v). (Compare this with the “action” of proj on the “standard basis”
n e 1 =
„ 1 0
« , e 2 =
„ 0 1
«o of R^2 .)
h) Find the coordinates of u, v, projL(u), and projL(v) with respect to the basis { u, v }. (Hint: For example, since u = 1 u + 0 v, the coordinates of u with respect to the basis {u, v} are simply (1, 0).)
i) Find the coordinates of e 1 , e 2 , proj(e 1 ), and proj(e 2 ) with respect to the basis { e 1 , e 2 }. (Compare with the answers of the preceding part.)
j) Since {u, v} is a basis for R^2 , every vector x ∈ R^2 can be written as x = μ u + ν v uniquely. Express projL(x) in terms of μ, ν, u, and v. Explain why projL is the orthogonal projection of R^2 onto the 1-dimensional subspace L ⊂ R^2. (Note also that u and v are orthogonal to each other.)
A little preview:
- The left-multiplication matrix» PL ∈ R^2 ×^2 you found for the projection map projL “looks” far more complicated than 1 0 0 0 - , and yet the “intrinsic geometry” of the two projection maps proj and projL are the same: projection onto some line. Whether that line is “horizontal” or not is really a consequence of our choice of coordinates, rather than the intrinsic geometric nature of the linear map. One way to regard this phenomenon is: The “standard coordinates” of R^2 , arising from the “standard basis”
n e 1 =
„ 1 0
« , e 2 =
„ 0 1
«o , is “ill-adapted” to the map projL. If we “transform” the standard basis to the basis {u = Rθ (e 1 ), v = Rθ (e 2 )} using the “transformation” Rθ , the resulting “coordinates” become “well-adapted” to projL in the sense that the “matrix representative” of projL with respect to these new coordinates takes the transparent form of (Rθ )−^1 · PL · Rθ =
» 1 0 0 0
- . The notions of eigenvectors, eigenvalues, diagonalization, etc. concern the determination of bases “well-adapted” to a given linear map so that the matrix representative takes on the simplest possible form, making the intrinsic geometry of the linear map transparent.
- Let L ⊂ R^2 be a 1 -dimensional subspace. For any x ∈ R^2 , its orthogonal projection projL(x) ∈ L into L is the unique vector in L which is closest to x. (Do you believe it? Can you visualize/explain it?) So, projL(x) is the “best approximation” of x within the subspace L. The high-dimensional analogue of this idea is widely employed in practice whenever an approximation process of some sort is involved. We will see some more of this under the topic of “data fitting.”
Recommended Exercises from Textbook (Do NOT hand in, but know these for tests/exam.) Section Problem(s) 2.1 1 – 13, 15, 16, 23
- Read §2.1 of the course text.
a) Let A =
, B =
, and C =
. Compute A · (B + C) and AB + AC. (Note that they are
equal.) b) Let
A =
a 11 a 12 a 13 a 21 a 22 a 23
∈ R^2 ×^3 and B =
b 11 b 12 b 21 b 22 b 31 b 32
5 , C =
c 11 c 12 c 21 c 22 c 31 c 32
5 ∈ R^3 ×^2.
Use the “Row-Column Rule” on p.111 of the course text (see also Example 5 on the same page) to compute the (1, 2)-entry of A(B + C) and the (1, 2)-entry of AB + AC. (Note that they are equal.) (Convince yourself that A(B + C) = AB + AC ∈ Rm×k^ , for any A ∈ Rm×n^ and B, C ∈ Rn×k^ .)
