Math 251 Exam 3

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

a set where elements can be added and multiplied by scalars so that certain conditions are satisfied; ex: all functions from R to R

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 matrix is a square matrix with numbers on the leading diagonal and zeros in all other places. b. identity or zero matrix c 2x2 matrix with all 1 entries

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

f(x)=M⋅x

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

if there are scalars c1,...,cn so that v=c1v1+...+cnvn.

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 ×2 v=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 .

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 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 ×2 v|=|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|.

Give a geometrical (coordinate free) definition of the double scalar product (determinant) for vectors on the plane. (and picture)

u→ (×2)v→ =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.

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

Give a geometrical (coordinate free) definition of the vector product.

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.

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. f 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. u⋅u=‖u‖^2

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

v≠0 and M⋅v=λ⋅v for some scalar λ.

Give a definition of an eigenvector of a matrix M

v≠0 and M⋅v=λ⋅v for some scalar λ.

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.

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. Picture

points P,Q,R,S lie on the same plane.

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

det(M− λ⋅I)

Define the identity 3 by 3 matrix.

[[1 0 0],[0 1 0], [0 0 1]]

Define the transpose of a matrix.

the rows of the transpose are the columns of the original

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.

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. Picture

Two vectors PQ→ and PR→ are linearly dependent if points P,Q,R lie on the same line.

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 MT equals the inverse M−1 of M.

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.

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.

AA^(T)=I,

Give a definition of a symmetric matrix M

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

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 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.

Explain the zero vector geometrically.

A vector AA→ is called zero vector, denoted 0→ or 0.

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.

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.

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.

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

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

The range of T•V→ is all vectors of the form T(v) for v in V. If T(v) =M•v, then the range is all B so that M•v = B has a solution

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).

Sketch a picture illustrating addition of vectors.

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

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 in space. Provide the answer first consisting of words only.

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.

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[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.

Describe the Gram-Schmidt algorithm

Given v1, v2,...: w1=v1; w2 = v2-p2 (p2 is projection of v2 and w); w3 = v3-p3 (p3 is projection of v3 onto spaces (w1,w2)

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.

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 ∥.

Give a definition of an orthogonal matrix M

M^-1 = M^T

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

a function f:V→W is a so that f(u+w)=f(u)+f(w) and f(k*w) = k*f(w) ex: derivative and the definite integral.

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

M•N = NM•I

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→.

Given two vectors u→ and v→ what do we mean by the parallel-perpendicular decomposition of v→?

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.

The area of the parallelogram spanned by u→ and v→ is ‖u→‖⋅‖v→‖⋅sin⁡φ where φ is the angle between u→ and v→ (recall, we consider the angle to be between 0 and 180 degrees).

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 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 negative of a geometric vector.

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

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.

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→.

Define vectors geometrically.

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 are added as follows: To get a→+b→, move the tail of b→ at the tip of a→; then a→+b→ goes from the tail of a→ to the tip of b→, i.e., AB→+BC→=AC→. For k>0, k times a vector a→, denoted ka→ has the same direction as a→, but k times its length. For k<0, ka→ has the opposite direction as a→, but |k| times its length.

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.

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

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.

Define the product of two matrices

element (i,j) of AB is rows of A dot columns of B

Derive (from the geometric definition) basic double scalar products u ×2 v, 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).

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.

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?

kernel of T•V→W is all v in V that T(v) = 0→; If T(v) =M•v, then the kernel's all v so that M•v= 0→

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

λ^2−(trM)λ+(detM)