MATH251 - Exam 3 Complete

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What does it mean that N is the inverse of a matrix M?

A matrix N is the inverse of M if M⋅N equals the identity matrix.

Explain equality of geometric vectors. Provide the answer first consisting of words only. Then draw a picture.

A nice geometrical criterion for two vectors AB→ and CD→ to be identical is the intersection of segments AD and CB to be their common midpoint.

Give a definition of a symmetric matrix M.

A square matrix M is called symmetric if MT=M.

Linear/Matrix Algebra

A study of simultaneous linear equations

Explain the zero vector geometrically.

A vector AA→ is called zero vector, denoted 0→ or boldface 0. It has no magnitude.

Define vectors algebraically.

Algebraically, vectors are any objects that can be added and multiplied by scalars so that the regular rules are satisfied (commutativity of addition, associativity of addition, associativity of multiplication, distributivity of multiplication with respect to addition). For example, all functions may be considered as vectors. Typically used algebraic vectors are arrays [a1,a2,...,an] and their addition [a1,a2,...,an]+[b1,b2,...,bn] is defined as [a1+b1,a2+b2,...,an+bn]. The multiplication of [a1,a2,...,an] by a scalar c is defined as [c⋅a1,c⋅a2,...,c⋅an].

Give a definition of a vector space. Give an example of a non-euclidean vector space.

Any set of objects V where addition and scalar multiplication are defined and satisfy properties 1--7 below is called a vector space. Here by addition we mean any operation which associates with each pair of objects A and B from V another object (the sum) C also from V; by a scalar multiplication we mean any operation which associates with every scalar k and every object A from V another object from V called the scalar multiple of A and denoted by kA. 1. The addition is commutative and associative: A+B=B+A, A+(B+C)=(A+B)+C 2. The multiplication by scalar is distributive with respect to the addition: a(B+C)=aB+aC 3. The product by a scalar is distributive with respect to the addition of scalars: (a+b)C=aC+bC 4. a(bC)=(ab)C 5. 1∗A=A (here 1 is the scalar 1) 6. There exists a zero vector 0 such that 0+A=A+0=A for every A 7. 0∗A=0 (here the first 0 is the scalar 0, the second 0 is the zero-vector) The basic example of a non-euclidean vector space is the space of real-valued functions f. The addition f+g is defined by (f+g)(x)=f(x)+g(x) The multiplication c⋅f by a scalar c is defined by (c⋅f)(x)=c⋅f(x)

Give a geometrical (coordinate free) definition of the double scalar product (determinant) for vectors on the plane. Provide the answer first consisting of words only. Then draw a picture.

Definition: u×(2)v is the number whose length equals the area of the parallelogram spanned by u and v and whose sign is determined by the right hand rule.

Give a geometrical (coordinate free) definition of the vector product. Provide the answer first consisting of words only. Then draw a picture.

Definition: u×v is the unique vector perpendicular to both u and v whose length equals the area of the parallelogram spanned by u and v and whose direction is determined by the right hand rule.

Triple scalar product

Denoted by the det(u→,v→,w→), where u→, v→, and w→ are 3-vectors. It is the number whose absolute value equals the volume of the parallelpiped spanned by u→,v→,w→ and whose sign is determined by the right hand rule.

Double scalar product

Double scalar product of u→ and v→ (where u→ and v→ are 2-vectors), denoted u→ X2 v→, is the number whose absolute value equals the area of the parallelegram spanned bu u→ and v→, and whose sign is determined by the right hand rule. Also the determinant of a 2x2 matrix

Explain magnitude and direction of geometric vectors. Explain multiplication of geometric vectors by real numbers. Provide the answer first consisting of words only. Then draw a picture.

Each vector v→ has its magnitude (also known as length or modulus) ||v|| and its direction. Both are associated with the ability of multiplying vectors by real numbers: For planar vectors one can identify the direction of v →with the angle from the x-axis to v→ in the counterclockwise direction. For general vectors v→ ≠0 their direction will be understood as the unit vector of v→ : (v→ / ||v||). Possible notations: unit(v→ ), dir(v→ ). If m>0, then m * v→ has the same direction as v→ but its magnitude is m*||v||. 0*v→ =0. m<0, then m*v→ =−(−m)*v→ .

Explain the angle between two vectors on the plane xy. Provide the answer first consisting of words only. Then draw a picture.

For vectors on the plane we have a very precise concept of the /_(u→ ,v→ ) from u→ to v→. It is measured from u→in the counterclockwise direction until we encounter the direction of v→. /_(u→,v→) is considered to be a number modulo 360 ( modulo 2π if measured in radians). For example: /_(u→,v→)=−90 means the direction of v→ is obtained from dir(u→) by clockwise rotation by 90 degrees.

Define the transpose of a matrix.

Given a matrix M one can find its transpose MT according to the following rule: the (i,j) entry of MT equals the (j,i) entry of M.

Explain the angle between two vectors in space. Provide the answer first consisting of words only. Then draw a picture.

Given two geometric vectors u→ and v→ we can loosely talk about the angle between them. There are two possible explanations/answers: one is the smaller angle, the other is the bigger angle. Both answers add up to 360 degrees. For general vectors we are only interested in cos(α), so it does not matter which angle of the two possible ones we choose as cos(α)=cos(360−α). For practical purposes we may agree that only angles from 0 to 180 degrees (0 to π if measured in radians) are of interest.

Explain the parallelogram spanned by two geometric vectors. Give a parametrization of the parallelogram spanned by vectors u→ and v→ whose initial point is the origin. Provide the answer first consisting of words only. Then draw a picture.

Given two geometric vectors u→ and v→ we can slide u→ along v→ and create the parallelogram spanned by u→ and v→. Algebraically, it means that we pick the endpoint of the vector t*u→, 0 ≤ t ≤ 1, and we add s*v→ for some 0 ≤ s ≤ 1. Thus the simplest parametrization of that parallelogram (in case the initial points of the vectors u→ and v→ is the origin) is: t*u→ + s*v→ where 0 ≤ s, t ≤ 1.

Define the product of two matrices.

Given two matrices M and N one can find their product P=M⋅N according to the following rule: the (i,j) entry of P is the dot product of i-th row of M and j-th column of N. Thus, for the product to be defined, the number of columns of M must be equal to the number of rows of N. The simplest case is R⋅C, the product of a row vector R and a column vector C of the same number of entries.

Give a definition of the kernel of a linear transformation of vector spaces. What is the connection of this definition to systems of linear equations?

If f:V→W is a linear transformation, then the kernel of f is {v∈V|f(v)=0}. If a system of homogeneous linear equations is converted to the matrix form, then it becomes M⋅X=0. The solution space of that system is identical to the kernel of the linear transformation f defined by f(X)=M⋅X and is called the null-space of M.

Give a definition of the range of a linear transformation of vector spaces. What is the connection of it to systems of linear equations?

If f:V→W is a linear transformation, then the range of f is {w∈W|w=f(v) for some v∈V}. If a system of non-homogeneous linear equations is converted to the matrix form, then it becomes M⋅X=B. The space of vectors B for which there is a solution X is identical to the range of the linear transformation f defined by f(X)=M⋅X.

Give basic triple scalar products det(u,v,w), where u, v, and w range over basic vectors i, j, and k.

If two vectors are identical, the determinant is 0. Thus det(i,i,j)=0 and so on. det(i,j,k)=1 and flipping two vectors changes the sign.

Explain the parallelepiped spanned by three geometric vectors. Give a parametrization of that parralelepiped in case the initial points of the vectors u→, v→, and w→ is the origin. Provide the answer first consisting of words only. Then draw a picture.

In case of three geometric vectors we can slide the parallelogram spanned by u→ and v→ along w→ and create the parallelepiped spanned by u→, v→ and w→. Algebraically, it means that we pick the endpoint of the vector t*u→, 0 ≤ t ≤ 1, and we add s*v→ + q*w→ for some 0 ≤s, q ≤ 1. Thus the simplest parametrization of that parallelepiped (in case the initial points of the vectors u→, v→, and w→ is the origin) is t*u→ + s*v→ + q*w→ where 0 ≤ s, t, q ≤ 1.

Give a geometrical interpretation of the determinant of a 2 by 2 matrix. Provide the answer first consisting of words only. Then draw a picture.

Interpretation: det[u ,v] is the number whose length equals the area of the parallelogram spanned by u and v and whose sign is determined by the right hand rule.

What does it mean to normalize a non-zero vector v→? Give an example in the 3-space.

Let v→ be a non-zero vector. To normalize v→ is to find the unique unit vector with the same direction as v→. This is done by multiplying v→ by the reciprocal of its length; the corresponding unit vector is given by u→ = v→ / ||v||. Example: Consider R3 and the vector v→ =[1,2,3]. The norm (length) is √(14). Normalizing, we obtain the unit vector u→ pointing in the same direction, namely u→ =[ 1/√(14),2/√(14),3/√(14) ].

Give a definition of an orthogonal matrix M.

M is orthogonal if its inverse and its transpose are equal (M−1=MT).

Give a definition of the matrix of a linear transformation f:Rn→Rm.

M is the matrix of f:Rn→Rm if f(x)=M⋅x for all x∈Rn.

List possible notations for vectors.

Notations for vectors are often boldface lowercase letters like a, or lowercase letters with arrows on top, like a→ . If points A, B are specified: AB→. Boldface a a→ (with arrow on top) AB→(with arrow on top)

Describe the Gram-Schmidt algorithm.

Suppose that we have a basis v1,...,vn of a Euclidean vector space V. The next procedure, called the Gram-Schmidt algorithm, produces an orthogonal basis w1,...,wn of V. Let w1=v1 The vector w2 appears in the parallel-perpendicular decomposition v2=x⋅v1+w2. Next, we can find w3 as v3−p3, where p3 is the orthogonal projection of v3 onto the plane spanned by w1 and w2. Continuing in this manner, we can get all vectors wi.

Given two vectors u and v what do we mean by the parallel-perpendicular decomposition of v? Provide the answer first consisting of words only. Then draw a picture.

That means expressing v as A+B, where A is parallel to u and B is perpendicular to u.

Explain how to find the area of the parallelogram spanned by non-zero vectors u and v using the angle between them. Provide the answer first consisting of words only. Then draw a picture.

The area of the parallelogram spanned by u and v is ||u||*||v||sinφ where φ is the angle between u and v (recall, we consider the angle to be between 0 and 180 degrees). Let φ be the angle between u and v (recall, we consider the angle to be between 0 and 180 degrees). Notice the height of the parallelogram spanned by u and v (if v is its base) is ||u||⋅sinφ Therefore the area of the parallelogram spanned by u and v is ||u||*||v||⋅sinφ

Define dot product of array vectors algebraically.

The dot product [a1,a2,...,an]⋅[b1,b2,...,bn] is defined as a1⋅b1+a2⋅b2+...+an⋅bn.

Define the identity 3 by 3 matrix.

The identity matrix In has its diagonal entries equal to 1 and off-diagonal entries are all 0.

Explain the negative of a geometric vector.

The negative of a vector corresponds to the same segment, but with opposite direction. Thus −PQ→= QP→.

Give a definition of the rank of a matrix M.

The rank of M is the dimension of the space spanned by its columns (equivalently, rows).

The scalar component of a projection

The scalar component of u in the direction of v is x-new, the x coordinate of the tip of u in the new coordinate system. Let θ be the angle from v to u. As in basic geometry, x-new = ||u||*cosθ and we can express it in the dot product: ||u||*cosθ = || i-new || * || u || * cosθ = u dot i-new = u * dir(v) = (u dot v) / ||v||

Sketch a picture illustrating addition of vectors.

The sum of vectors is the diagonal of the resulting parallelogram.

Give a definition of the trace of a square matrix M.

The trace of a square matrix M is the sum of all elements aii on its diagonal.

State the form of the characteristic polynomial of a 2 by 2 matrix that uses the trace of M.

Theorem: For a 2×2 matrix M, the characteristic polynomial is λ2−(trM)λ+(detM).

Define the scalar component of a vector u with respect to vector v. Provide the answer first consisting of words only. Then draw a picture.

Think of v as pointing in the direction of the new x-axis. The new i-vector is inew=dir(v)=v / ||v|| and the new j-vector, jnew, is on the plane spanned by v and u. The scalar component of u in the direction of v is xnew, the x-coordinate of the tip of u in the new coordinate system. Let θ be the angle from v to u. As in basic geometry, xnew=||u|| cosθ and we can express it using the dot product: ||u|| cosθ=|| inew ||⋅||u||cosθ=u⋅inew=u⋅dir(v=(u⋅v)/|v|.

Define the vector component of a vector u with respect to vector v. Provide the answer first consisting of words only. Then draw a picture.

Think of v as pointing in the direction of the new x-axis. The new i-vector is inew=dir(v)=v / ||v|| and the new j-vector, jnew, is on the plane spanned by v and u. The vector component of u in the direction of v is xnew⋅inew, the orthogonal projection of u onto v. Let θ be the angle from v to u. As in basic geometry, xnew=||u||cosθ and we can express it using the dot product: ||u||cosθ=|| inew ||⋅||u||cosθ=u⋅inew=u⋅dir(v=(u⋅v)/|v|, so the vector component of u in the direction of v is ((u⋅v)/|v|2)⋅v.

Give a geometric meaning of the fact that three vectors PQ→, PR→, and PS→ are linearly dependent. Here P,Q,R,S are points in the 3-space R3.

Three vectors PQ→, PR→, and PS→ are linearly dependent if points P,Q,R,S lie on the same plane. Here P,Q,R,S are points in the 3-space R3.

Define orthogonal projection of a vector v onto vector u. Provide the answer first consisting of words only. Then draw a picture.

To project vector v orthogonally onto vector u means to find vector A parallel to u such that B :=v −A is perpendicular to u.

Orthogonal Projection

To project vector v→ orthogonally onto vector u→ means to find vector A→ parallel to u→ such that B→ := v→ - A→ is perpendicular to u→.

Give a geometric meaning of the fact that two vectors PQ→ and PR→ are linearly dependent. Here P,Q,R are points in the 3-space R3.

Two vectors PQ→ and PR→ are linearly dependent if points P,Q,R lie on the same line. Here P,Q,R are points in the 3-space R3.

Vector v is a linear combination of v1,...,vn. What does that mean?

Vector v is a linear combination of v1,...,vn if there are scalars c1,...,cn so that v=c1v1+...+cnvn.

Inconsistent vectors

Vectors are inconsistent if their system of equations is false

Define vectors geometrically. Provide the answer first consisting of words only. Then draw a picture.

Vectors can be viewed as directed line segments (arrows) between two points in the plane (or in space), but with the stipulation that translating such segments (without turning) results in the same vector. Each vector v has its magnitude (also known as length or modulus) ||v|| and its direction. Vectors can be added, and multiplied with (real) numbers. Numbers are called scalars when used in the context of vectors.

Vectors v1,...,vn form a basis of a vector space V. What does that mean?

Vectors v1,...,vn form a basis of V iff they are linearly independent and all vectors of V are linear combinations of vectors v1,...,vn.

a. Give a definition of diagonal 2x2 matrices. b. Give an example of a diagonal 2x2 matrix. c. Give an example of a 2x2 matrix that is not diagonal.

a. A diagonal 2x2 matrix is one with all entries off the diagonal equal 0. b. Identity or zero matrix is a diagonal 2x2 matrix. c. The 2x2 matrix with all entries 1 is not diagonal.

a. Give a definition of a unit vector. b. Give an example of a unit vector. c. Give an example of a vector that is not a unit vector.

a. A unit vector is a unit-length element of Euclidean space. Equivalently, one may say that the norm of a unit vector is equal to 1, and write ||u|| =1, where u→ is the vector in question. b. c.

a. Give a definition of orthogonal 2x2 matrices. b. Give an example of an orthogonal 2x2 matrix. c. Give an example of a 2x2 matrix that is not orthogonal.

a. An orthogonal 2x2 matrix is any U whose transpose UT is the inverse of U. b. Identity matrix is an orthogonal 2x2 matrix. c. The 2x2 matrix with all entries 1 is not orthogonal.

Give a definition of the characteristic polynomial of a square matrix M.

det(M−x⋅I) is a polynomial with the unknown x (usually denoted by λ). This polynomial is called the characteristic polynomial of the matrix M.

Explain the connection of the 3×3 determinant to cross product.

det(u,v,w)=(u×v)⋅w=u⋅(v×w)=v⋅(w×u). The height of the parallelepiped spanned by u, v, and w (assuming the base is the parallelogram spanned by u and v) is the projection of w onto u×v. That means it corresponds to (u×v)⋅w / |u×v| The volume corresponds to the height times the area of the base which is |u×v|, so the final answer is det(u,v,w)=(u×v)⋅w More generally, det(u,v,w)=(u×v)⋅w=u⋅(v×w)=v⋅(w×u).

Define algebraically the determinant of the matrix [[a11,a12],[a21,a22]].

det[[a11,a12],[a21,a22]]=a11⋅a22−a12⋅a21 .

Give a geometrical (coordinate free) definition of the triple scalar product. Provide the answer first consisting of words only. Then draw a picture.

det[u ,v ,w] is the number whose length equals the volume of the parallelepiped spanned by u, v and w and whose sign is determined by the right hand rule.

Give a geometrical interpretation of the determinant of a 3 by 3 matrix. Provide the answer first consisting of words only. Then draw a picture.

det[u,v,w] is the number whose length equals the volume of the parallelepiped spanned by u, v and w and whose sign is determined by the right hand rule.

Give a definition of a linear transformation of vector spaces. What are two basic examples of linear transformations from calculus?

f:W→V is a linear transformation if V,W are vector spaces and f(a⋅v+b⋅w)=a⋅f(v)+b⋅f(w) for all a,b∈R and all v,w∈W. Basic linear transformations from calculus are the derivative and the definite integral.

Derive (from the geometric definition) basic double scalar products u×2v, where u and v range over basic vectors i and j.

i×2i=j×2j=0 (the parallelogram has no area), i×2j=1 (the parallelogram is the square of area 1 and vectors are positively oriented), j×2i=−1 (the parallelogram is the square of area 1 and vectors are negatively oriented).

Derive (from the geometric definition) basic cross products u×v, where u and v range over basic vectors i, j, and k.

i×i=j×j=k×k=0 (no area), i×j=k (because k is perpendicular to i,j and the parallelogram has area 1), j×i=−k (the sign comes from the right hand rule), i×k=−j, k×i=j, j×k=i, k×j=−i.

Explain how to detect if non-zero vectors u and v are parallel using double scalar product.

u is parallel to v if only if the parallelogram spanned by them has area 0. Thus, the basic equation for two vectors being parallel is u×2v=0.

Explain how to detect if non-zero vectors u and v are parallel using cross product.

u is parallel to v if only if the parallelogram spanned by them has area 0. Thus, the basic equation for two vectors being parallel is u×v=0(vector) .

Explain how to detect if non-zero vectors u→ and v→ are parallel using dot product.

u is parallel to v only if the angle α between them is 0 or 180 degrees. Also, |cos(α)|=1 if and only if α=0 or α=180 (we assume 0≤α≤180). Thus, the basic equation for two vectors being parallel is |cos(α)|=1. It is equivalent to |u|⋅|v|⋅|cos(α)|=|u|⋅|v| or |u⋅v|=|u|⋅|v|.

Explain how to detect if non-zero vectors u→ and v→ are perpendicular (or orthogonal) using dot product.

u is perpendicular to v (u⊥v) iff u⋅v=0. Indeed, the vectors are perpendicular if and only if the angle α between them is 90 degrees. Also, cos(α)=0 if and only if α=90 (we assume 0≤α≤180). Thus, the basic equation for two vectors being perpendicular is cos(α)=0. It is equivalent to |u|⋅|v|⋅cos(α)=0 or u⋅v=0.

Explain how to detect if non-zero vectors u and v are perpendicular (or orthogonal) using double scalar product.

u is perpendicular to v if only if the parallelogram spanned by them is a rectangle. A parallelogram spanned by u and v is a rectangle if and only if its area is the product of magnitudes of u and v. Thus, the basic equation for two vectors being perpendicular is |u×2v|=|u|⋅|v|.

Explain how to detect if non-zero vectors u and v are perpendicular (or orthogonal) using cross product.

u is perpendicular to v if only if the parallelogram spanned by them is a rectangle. A parallelogram spanned by u and v is a rectangle if and only if its area is the product of magnitudes of u and v. Thus, the basic equation for two vectors being perpendicular is |u×v|=|u|⋅|v|.

Explain how to detect if non-zero vectors u, v, and w are coplanar using the triple scalar product.

u, v, and w are coplanar if only if the parallelepiped spanned by them has volume 0. Thus, the basic equation for three vectors being coplanar is det(u,v,w)=0.

Vector Product/Cross Product

u→ X v→ is the unique vector perpendicular to both u→ and v→ whose length equals the area of the parallelogram spanned by u→ and v→ whose direction is determined by the right hand rule.

Give a geometrical (coordinate free) definition of dot product.

u→⋅ v→ is the product of their lengths and the cosine of the angle between them. If u and v are unit vectors, then their dot product u⋅v is the cosine of the angle between u and v. In particular, i⋅i=1, i⋅j=0, and so on. The special case, u⋅u, of scalar product of u with itself, is the scalar square of the vector u, In Rn it equals to the square of the length of u: u⋅u = ||u||^2

Give a definition of an eigenvector of a matrix M.

v is an eigenvector of M if v≠0 and M⋅v=λ⋅v for some scalar λ.

Vectors v1,...,vn are linearly independent. What does that mean?

v1,...,vn are linearly independent iff c1v1+...+cnvn=0 implies c1=...=cn=0.

Give a definition of an eigenvalue of a matrix M.

λ is an eigenvalue of M if M⋅v=λ⋅v for some vector v≠0.