5- Matrix Algebra

Question 1

What’s a vector equation?

What does it mean “checking whether a vector can be written as a linear combination of vectors?
What does it mean that there is a solution?

Answer 1

• It is a linear combination of vectors.
• is the same as solving it as a linear system;
• if the system as a solution then vector b can be generated as a linear combination of vectors.

Question 2

Matrix:
- what is it?
- what is aij?
- main diagonal
- diagonal matrix
-identity matrix
- zero matrix
- matrix addition
- scalar multiplication

Answer 2

• it is a 2 dimensional array;
• entries aij where i=j;
• matrix whose non-diagonal entries are zero;
• it is a diagonal matrix where all diagonal entries are ones;
• matrix with all entries equal to 0;
• sum of the corresponding entries of the matrices;
• each entry of the matrix is multiplied by a scalar value.

Question 3

Matrix-Vector Multiplication:
- notation
- requirement
- what is it
- when does it have a solution?

Answer 3

• Ax = v
• n cols matrix A = n rows vector x
• it is a linear combination of vectors (represented by the columns of the matrix) with the scalar multiples in the vector x.
• it has a solution if and only if b is a linear combination of the cols of matrix A.

Question 4

Write the notation of the matrix equation, vector equation and associated augmented matrix… what do they have in common?

Answer 4

• Ax = v
• x1a1 + … + xnan = b
• [a1 … an b]
• they have the same solution set

Question 5

Ax = b:
- we can think that the matrix A is…
- we are talking about a linear transformation if… (2 requirements)
- x in R^… into b in R^…

Answer 5

• as an object that transforms vector x to produce a new vector b.
• if before and after the transformation, the origin is the same and vectors are linear.
• x in R^n is transformed into x in R^m.

Question 6

Linear Transformation properties:
- Additivity
- Scalar multiplication

Answer 6

• T(u + v) = T(u) + t(v)
• T(r * v)= r*(T) with r that is a scalar.

Question 7

Given 2 matrices Amxn and Bnxp then:
- what is the product AB? What dimensions?

Answer 7

It is the matrix mxp whose columns are Ab1, Ab2… Abp, that is A * [b1 b2 … bp] = [Ab1 Ab2 … Abp]

Question 8

Matrix Multiplication Properties:
- Associative law of multiplication
- Left distributive law;
- Right distributive law;
- for any scalar r;
- identity for matrix multiplication

Warnings:
- in general …
- in general…
- in general…

Answer 8

• A(BC) = (AB)C;
• A(B+C) = AB + AC;
• (A+B)C = AC + BC;
• r(AB) = (rA)B = A(rB);
• I A = A = A I.

Warnings:
- AB != BA;
- if AB = AC then is not true in general that B=C;
- if AB= 0 then is not true in general that either A or B is equal to the zero matrix.