Vectors and Units Flashcards
What shape does the set of all the vectors created by linear combinations of non-collinear vectors A and B, create? What is this set called?
If vectors A and B are not collinear, the set creates a plane (else, it’ll be a line), so the whole R2 set. you can represent any vector in R2 using a linear combination of vectors A and B.
This set is called the span of the vectors A and B. Span(A,B)=R2
What are the two most familiar vectors that span R2?
[0,1] and [1,0], these form a basis for R2
What does linear dependency mean? 2 definitions
.
It means that one of the vectors in the set, can be represented by a linear combination of the other vectors in the set. in other words: the only value for “c”s so that the below equation holds, is not 0 c1V1+c2V2+…+cnVn= 0
Under what conditions is subset V a -linear- subspace of Rn?
Linear subspaces 02:19
subset V is a subspace of Rn, if:
1) V contains zero vector
2) if vector X in the subset V, then cX has to be in it too (closure under scalar multiplication)
3) If vectors A and B are in the subset V, then A+B must be in subset V too (closure under addition)
The span of the set of any number of n -size- vectors, is a valid subspace of Rn. True/False why?
Linear subspaces 23:23
True
The span is the set of all the linear combinations of the vectors in the set 15:08.
Using 0 as the coefficients of the linear combinations we achieve condition 1: including zero vector
Condition 2: closure under scalar multiplication: a(c1V1+c2V2+c3V3) is still going to be a linear combination of the vectors and is still going to be in the span
Condition 3: closure under addition
c1V1+c2V2+c3V3 + c4V1+c5V2+c6V3 is still a linear combination of the three n -size- vectors in the set, so it’ll be in the span too
By satisfying all three conditions, we conclude that the statement is true
What is a basis for a subspace?
Basis of a subspace 02:54
we said that the span of a subset of n vectors creates a subspace in Rn. if the vectors in the subset are linearly independent, then these vectors are a basis for that subspace
Basis: linearly independent vectors whose linear combination can reach any point in the subspace
How can we parametrically represent vectors in 2D and higher spaces, given 2 linearly independent vectors A and B?
{A+t(B-A)|t∈R}
or
{B+t(B-A)|t∈R}
or
{A+t(A-B)|t∈R}
or
{B+t(A-B)|t∈R}
is the below set, a subspace of Rn?
{x∈Rn|Ax=0}
X: a vector A: a matrix m * n
Yes, because it satisfies the 3 conditions:
1) the set contains 0
2) closure under addition
3) closure under multiplication by scalar
This subspace is called the null space of A, denoted by N(A), so null space of A is all the vectors that satisfy Ax=0
Note: A(x+y)=Ax+Ay for matrices (incl. vectors)
Under what condition are column vectors of A, linearly independent?
Hint: A has a null subspace
we have: {x∈Rn|Ax=0} for null subspace, based on the video this happens if and only if the only solution for the below equation:
x1V1+x2V2+…+xnVn=0
is if x1=x2=…=xn=0
So this happens if and only if (this means that it’s true both ways), N(A) contains only the zero vector, and if the column vectors of A are linearly independent N(A) contains only the zero vector
How is the column subspace of matrix A defined?
Column space of a matrix
01:08
Column subspace is all the linear combinations of A’s column vectors, aka. the span of the column vectors’ set. It’s shown by C(A)
Note: C(A)={x1V1+x2V2+…+xnVn|x1,…,xn ∈R}=span(V1,V2,…,Vn). so if vector b not in C(A) then Ax=b won’t have a solution because Ax=x1V1+x2V2+…+xnVn, where Vs are column vectors of A
What number is the dimension of a subspace?
Number of the elements in a basis set for the subspace.
Note:
In general, the nullity (dimension of the null space, aka. # elements in a basis set of the null space) of any matrix, is number of non-pivot (free) columns in row-reduced echelon form (13:53)
Null Space Calculator the yellow rows, show the non-pivot columns (linearly dependent columns)
What is the dimension of the column space called?
Rank of the matrix. (it’s the number of linearly independent columns of the matrix, or the number of elements in the basis of the column space)
Dimension of the column space of a matrix is called ____ and is the number of ____ which is equal to the ____ columns of the matrix.
Dimension of the null space of a matrix is called ____ and is the number of ____ which is equal to the ____ columns of the matrix.
rank, elements in a basis set for column space of a matrix, pivot
nullity, elements in a basis set for null space of a matrix, non-pivot
