Basis and Dimension Flashcards
What is a basis and what is the invariance of dimension?
A list of vectors in space V that is linearly independent and spans V.
If B and C are both vectors of V, then they contain the same number elements
When is a vector space finite dimensional and what is its dimensions?
If there exists a basis B for V with a finite number of elements.
Dimension is |B|.
What is the dimension of the zero vector space?
Dimension is 0.
What is the dimension of the subspace of V?
If V finite-dimensional, then the subspace W is also finite-dimensional.
Dim(W) <= Dim(V). If equal, then W = V
What is the sifting algorithm?
Consider each vector consecutively. If vi is the zero vector or if it is a linear combination of other preceding vectors, then remove
How do we create a basis from a spanning list of vectors?
- Sifting the list
How do we create a basis from a list of vectors in a finite dimensional vector space?
If the list is linearly independent, then extend it to the basis
What does it mean if a list of n independent vectors is in an n-dimensonial vector space?
It is a basis
What is the coordinate vector with respect to a basis?
Writing v as a linear combination of the vectors in the basis, the scalars in the combination are the coordinates of v with respect to the basis B
How do you calculate the coordinate vector of a vector with respect to multiple bases and what does the answer mean?
Write the vector as a linear combination of each basis.
The coordinate vector is essentially then the amount of each basis vector that is needed to reconstruct the original vector.
How do we compute the coordinate vector of a vector with respect to basis C, if we know the coordinate vector of w with respect to basis B?
Express each vector in B as linear comb of basis vectors in C
What is the change of basis matrix?
P(C <- B) = Matrix of cordiante vectors of Bi with respect to C
Therefore the coordinate vectors of the current matrix with respect to the matrix being changed to
What is the formula for the coordinate vector of v ito a basis, using the COB matrix?
coordinate vector of v ito the basis changed to = C.O.B matrix * coordinate vector changed from.
Therefore [v]B = PB <- B’ * [V]B’
How do you calculate a basis for the solution set of a homogeneous system of linear equations?
- Write down augmented matrix
- Perform row operations into RREF
- Identify pivot columns and free variables
- Write general solutions, expressing the variables in terms of the free variables
- Construct the basis vector which is the vector the free variables correspond to
