Matrix Algebra Definitions

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. Words and 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 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 ∥u⃗ ∥=1, where u⃗ is the vector in question.

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.

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. We can slide the parallelogram spanned by u⃗ and v⃗ along w⃗ and create the parallelepiped spanned by u⃗ , v⃗ and w⃗ .

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.

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.

Define vectors algebraically.

Algebraically, vectors are any objects that can be added and multiplied by scalars so that the regular rules are satisfied (associativity of addition, associativity of multiplication, distributivity of multiplication with respect to addition)

Explain the angle between two vectors in space.

Angle measured from one vector counterclockwise to the other vector.

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 EX. space of real-valued functions f.

Explain the angle between two vectors on the plane xy

For vectors on the plane we have a very precise concept of the ∠(u⃗ ,v⃗ ) from u⃗ to v⃗ . 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.

Define the identity 3 by 3 matrix

I3 has its diagonal entries equal to 1 and off-diagonal entries are all 0.

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.

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.

Explain magnitude and direction of geometric vectors. Explain multiplication of geometric vectors by scalars.

Magnitude is the length of the vector. For planar vectors one can identify the direction of v⃗ with the 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 ). Multiplying by a positive scalar increases the magnitude by the scalar. If the scalar is negative the same is true, but the direction is switched.

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

Negative of a vector goes in the opposite direction of it's positive counterpart.

List possible notations for vectors.

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

Define orthogonal projection of a vector v⃗ onto vector u

Parallel decomposition v = A+B where A is parallel to u and B is perpendicular to u

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

Given two vectors u⃗ and v⃗ what do we mean by the parallel-perpendicular decomposition of v⃗ ?

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

The area of the parallelogram spanned by u⃗ and v⃗ is ∥u⃗ ∥⋅∥v⃗ ∥⋅sinφ where φ is the angle between u⃗ and v height = ∥u⃗ ∥⋅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

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).

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

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 using the dot product: ∥u∥cosθ=∥i_new∥⋅∥u∥cosθ=u⋅i_new=u⋅dir(v)=(u⋅v)/|v|.

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

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.

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

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).

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 v. The corresponding unit vector is given by u⃗ =v⃗ / ∥v⃗ ||. 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)

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

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

Vectors are perpendicular if and only if the angle α between them is 90 degrees. Formulae: cos(a) = 0 |u|*|v|*cos(a) = 0 u*v = 0

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 v⃗ has its magnitude (also known as length or modulus) ∥v⃗ ∥ and its direction.

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

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

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 3 by 3 matrix.

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 (coordinate free) definition of the triple scalar product.

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 interpretation of the determinant of a 2 by 2 matrix.

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.

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)

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 Hint: just draw out the expanded det() to get the signs.

Define the transpose of a matrix

the (i,j) entry of MT equals the (j,i) entry of M

Define the product of two matrices

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.

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

u is parallel to v if only if the parallelogram spanned by them has area 0 Or u(×2)v=0

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

u is parallel to v if only if the parallelogram spanned by them has area 0. u×v=0⃗ .

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. Formulae: |cos(α)|=1 |u|⋅|v|⋅|cos(α)|=|u|⋅|v| |u⋅v|=|u|⋅|v|.

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

u is perpendicular to v if only if the parallelogram spanned by them is a rectangle |u×v|=|u|⋅|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 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 |u(×2)v|=|u|⋅|v|

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. det(u⃗ ,v⃗ ,w⃗ )=0

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

u⃗ ×2v⃗ =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 geometrical (coordinate free) definition of the vector product.

u⃗ ×v⃗ is the unique vector perpendicular to both u⃗ and v⃗ whose length equals the area of the parallelogram spanned by u⃗ and v⃗ and whose 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.

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

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