Vectors and Spaces Flashcards
Learn everything there is to know in vectors and spaces
What is the parametric representation of a line?
L={x + tv | t ⋲ R}
What is a linear combination?
Combination(sum) of multiple vectors that have been scaled by arbitrary constants.
What is the Span{} of a space?
It is the sett for all vectors whose linear combination can make up the whole space.
What is linear independence?
A set of vectors is linearly independent iff the vectors can’t be created by the linear combination of the other vectors in the set.
A set is linearly independent iff it only has the trivial solution.
What is a linear subspace?
V is a subset of ℝ
V is a subspace of ℝ:
-V contains 0
-If x ⋲ V then cx | c⋲ ℝ –> V. Closure under
multiplication
-Closure under addition. a⋲ V and b⋲ V then
(a + b)⋲ V.
The span of all the LI vectors is a valid subspace.
What is the basis of a subspace?
A set of vectors is a basis of V iff those vectors form a LI span of the subspace V. The basis is the minimum set of vectors that span V.
What is the dot product?
The dot product is the square of the length of the vector in the same direction. a.b =|a||b|cos(Θ).
a⟂b if a.b=0
What is the cross product?
|axb|=|a||b|sin(Θ).
a||b if axb=0.
Matrix vector products?
Ax only valid iff the vector x has the same amount of components as n in A mxn.
What is the null space of a matrix?
Nul(A) = Nul(rref(A)).
Nul(A)={x⋲ ℝ | Ax=0}
Nul(A) is a valid subspace of A for Ax=0. A set of vectors is LI iff Nul(A)={0}.
What is the column space of a matrix?
The column space of a matrix is the set of vectors that span said matrix.
Explain the null space and the column space basis.
The null space is the vectors that define span Ax=0.
The column space is the set of vectors that span A.
The basis of the column space is the minimum set of vectors that span the matrix. They are the set of the pivot columns.
What is the dimension of a Null space or nullity?
The dimension of a Null space is the number of elements that comprise the basis of the Null space.
How could you find the basis of a matrix/subspace?
We know that the basis is the minimum set of vectors that span said matrix or subspace. In the case of the matrix the basis is comprised of the vectors situated in the pivot columns.
What is the dimension(rank) of the columns space?
The dimension is the number of elements that make up the set of Linearly Independent vectors comprising he basis of the col(A).
