MATRIX 251
Give a definition of the characteristic polynomial of a square matrix M.
Definition: 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
Give a definition of a linear transformation of vector spaces. What are two basic examples of linear transformations from calculus?
Definition: 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.
Give a geometrical (coordinate free) definition of the vector product.
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.
Vectors v1,...,vn are linearly independent. What does that mean?
Definition: v1,...,vn are linearly independent iff c1v1+...+cnvn=0 implies c1=...=cn=0
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.
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→ 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|.
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 dot product
u→⋅v→ 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.
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 definition of the matrix of a linear transformation f:Rn→Rm
Definition: M is the matrix of f:Rn→Rm if f(x)=M⋅x for all x∈Rn.
Give a definition of the rank of a matrix M
Definition: The rank of M is the dimension of the space spanned by its columns (equivalently, rows).
Give a definition of an orthogonal matrix M
Definition: M is orthogonal if its inverse and its transpose are equal (M-1=MT)
Give a definition of a symmetric matrix M
A square matrix M is called symmetric if MT=M
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]
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 Diagonal matrix is a square matrix with numbers on the leading diagonal and zeros in all other places.
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.
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.
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 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 AA and denoted by kA.
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→AA→ is called zero vector, denoted 0⃗ 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
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.
Two vectors PQ→ and PR→ are linearly dependent if points P,Q,R lie on the same line. Here P,Q,RP,Q,R are points in the 3-space R3.
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→∣. 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.
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.
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 MM and j-th column of N. Thus, for the product to be defined, the number of columns of MM must be equal to the number of rows of N. The simplest case is R⋅C, the product of a row vector RR 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 ff 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 ff 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.
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 ∥.
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→.
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→?
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→‖⋅sinφ 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.
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 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→−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.
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
Sketch a picture illustrating addition of vectors.
To draw z1+z2 slide vector z2 on vector z1. The sum of vectors is the diagonal of the resulting parallelogram.
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→.
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.
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.
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 an eigenvalue of a matrix M
λ is an eigenvalue of M if M⋅v=λ⋅v for some vector v≠0.