M251 Quiz 9 and Exam 3

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 unit-length element of Euclidean space -the norm of a unit vector is equal to 1, and write ||u⃗|| = 1

Define vectors geometrically

-directed line segments between two points in the plane -translating such segments (without turning) results in the same vector -Each vector v⃗ has its magnitude ∥v⃗ ∥ and its direction. -Vectors can be added, and multiplied with (real) numbers -Numbers called scalars in context of vectors. Vectors are added as follows: To get a⃗ +b⃗ , move the tail of b⃗ at the tip of a⃗ ; then a⃗ +b⃗ goes from the tail of a⃗ to the tip of b⃗ , i.e., AB→+BC→=AC→ -For k>0, k times a vector a⃗ , denoted ka⃗ has the same direction as a⃗ , but k times its length. For k<0, ka⃗ has the opposite direction as a⃗ , but |k| times its length.

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

A matrix M is orthogonal if its transpose M^T equals the inverse M^−1 of M.

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.

Give a definition of a symmetric matrix M.

A square matrix M is called symmetric if M^T=M.

Define vectors algebraically

Algebraically, vectors are any objects that can be added and multiplied by scalars so that the regular rules are satisfied

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

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

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

Definition: det[u⃗ ,v⃗ ,w⃗ ] is the number whose length equals the volume of the parallelepiped spanned by u⃗ , v⃗ and w⃗ and whose sign is determined by the right hand rule.

Give a geometrical 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: u⃗ ×2 v⃗ is the number whose length equals the area of the parallelogram spanned by u⃗ and v⃗ and whose sign is determined by the right hand rule.

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 v⃗ has its magnitude ∥v⃗ ∥ and its direction. Both are associated with the ability of multiplying vectors by real numbers For planar vectors angle from the x-axis to v⃗ in the counterclockwise direction. For general vectors v⃗ ≠0 their direction will be understood as the unit vector of v⃗ : (v⃗ / |v⃗|) Possible notations: unit(v⃗ ), dir(v⃗ ). If m>0, then m⋅v⃗ has the same direction as v⃗ but its magnitude is m⋅||v⃗||. 0⋅v⃗ =0. m<0, then m⋅v⃗ =−(−m)⋅v⃗ .

State the form of the characteristic polynomial of a 2x2 matrix that uses the trace of M.

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

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

For general vectors we are only interested in cos(α) only angles from 0 to 180 degrees (0 to π if measured in radians) are of interest

Define the transpose of a matrix.

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

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=0⃗ . 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 det(i,i,j)=0 . det(i,j,k)=1 and flipping two vectors changes the sign

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.

Interpretation: det[u⃗ ,v⃗ ,w⃗ ] is the number whose length equals the volume of the parallelepiped spanned by u⃗ , v⃗ and w⃗ and whose sign is determined by the right hand rule.

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

It is measured from u⃗ in the counterclockwise direction until we encounter the direction of v⃗ . ∠(u⃗ ,v⃗ ) is considered to be a number modulo 360 ( modulo 2π if measured in radians). For example: ∠(u⃗ ,v⃗ )=−90 means the direction of v⃗ is obtained from dir(u⃗ ) by clockwise rotation by 90 degrees.

Give a definition of an orthogonal matrix M.

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

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 a⃗ . If points A, B are specified: AB→

Define the product of two matrices.

P = M⋅N (i,j) entry of P is the dot product of i-th row of M and j-th column of N. 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.

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 u⃗ and v⃗ what do we mean by the parallel-perpendicular decomposition of v⃗ ? Provide the answer first consisting of words only. Then draw a picture.

That means expressing v⃗ as A⃗ + B⃗ , where A⃗ is parallel to u⃗ and B⃗ is perpendicular to u⃗

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

The area of the parallelogram spanned by u⃗ and v⃗ is ||u⃗||⋅||v⃗||⋅sinφ where φ is the angle between u⃗ and v⃗

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.

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

The height of the parallelepiped spanned by u, v, and w is the projection of w onto 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 the identity 3 by 3 matrix.

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

Explain the negative of a geometric vector. Explain the zero vector geometrically.

The negative of a vector corresponds to the same segment, but with opposite direction. Thus −PQ→=QP→. A vector AA→ is called zero vector, denoted 0⃗ or 0.

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).

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.

Define the scalar component of a vector u⃗ with respect to vector v⃗ . 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 u⃗ with respect to vector v⃗ . 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.

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

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

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

To project vector v⃗ orthogonally onto vector u⃗ means to find vector A⃗ parallel to u⃗ such that B⃗ :=v⃗ −A⃗ is perpendicular to u⃗

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.

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.

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

a space consisting of vectors, together with the associative and commutative operation of addition of vectors, and the associative and distributive operation of multiplication of vectors by scalars. 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)

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 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 U^T = U^-1. b. Identity 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)

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

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

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.

det[u⃗ ,v⃗ ] is the number whose length equals the area of the parallelogram spanned by u⃗ and v⃗ 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.

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

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

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 j × i = −k i × k = −j k × i = j j × k = i k × j = −i.

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

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

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

u is parallel to v iff the parallelogram spanned by them has area 0

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

u is parallel to v iff the parallelogram spanned by them has area 0

Explain how to detect if non-zero vectors u⃗ and v⃗ 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

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

u is perpendicular to v iff the parallelogram spanned by them is a rectangle A parallelogram spanned by u and v is a rectangle iff its area is the product of magnitudes of u and v.

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

u is perpendicular to v iff the parallelogram spanned by them is a rectangle. A parallelogram spanned by u and v is a rectangle iff its area is the product of magnitudes of u and v.

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

u is perpendicular to v iff u⋅v = 0. Indeed, the vectors are perpendicular if and only if the angle α between them is 90 degrees.

Explain how to detect if non-zero vectors u⃗ , v⃗ , and w⃗ are coplanar using the triple scalar product.

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

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

u⃗ ×v⃗ is the unique vector perpendicular to both u⃗ and v⃗ length equals the area of the parallelogram spanned by u⃗ and v⃗ direction is determined by the right hand rule.

Give a geometrical (coordinate free) definition of dot product

u⃗⋅v⃗ is the product of their lengths and the cosine of the angle between them.

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.

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

we can slide the parallelogram spanned by u⃗ and v⃗ along w⃗ and create the parallelepiped spanned by u⃗ , v⃗ and w⃗ . Algebraically, it means that we pick the endpoint of the vector t⋅u⃗ , 0≤t≤1, and we add s⋅v⃗ +q⋅w⃗ for some 0≤s,q≤1. Thus the simplest parametrization of that parallelepiped (in case the initial points of the vectors u⃗ , v⃗ , and w⃗ is the origin) is t⋅u⃗ +s⋅v⃗ +q⋅w⃗ where 0≤s,t,q≤1.

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

we can slide u⃗ along v⃗ and create the parallelogram spanned by u⃗ and v⃗ Algebraically, it means that we pick the endpoint of the vector t⋅u⃗ , 0≤t≤1, and we add s⋅v⃗ for some 0≤s≤1. Thus the simplest parametrization of that parallelogram (in case the initial points of the vectors u⃗ and v⃗ is the origin) is t⋅u⃗ +s⋅v⃗ where 0≤s,t≤1.

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.