Ch3 Span, Kernel, Image & Subspaces Flashcards
What is the Kernel of a Matrix ?
The parts of the matrix which are always mapped to 0, this is essentially a set of coefficient vectors in a matrix which map to 0 when multiplied by the vector
How to find the Kernel of a Matrix ?
Solve the matrix down to RREF = 0 , identify the open variables and define our x1,x2 and x,3 values. then set up the matrix with respect to the two open variables.
How to solve for the Span of the Kernel of this matrix
Solve this matrix into RREF = 0, since all points in this matrix are defined our only Kernel is the 0 Vector
Ker(a) = {0}
We were able to reduce a matrix down to this RREF, How do we find the Span of the Kernel of this matrix ?
The Span of the Kernel is the set of Vectors which would create an infinite number of solutions: we find this by making x3 = t (an arbitrary variable) and solve the rest of this matrix with respect to t.
We find that since there is only one open variable our Span of the Kernel will be in R¹. and look like:
We are able to solve a n x m matrix down to RREF however it is a complete 0 vector, what is our Kernel ?
e₁, e₂, e₃ ——– e_n
Since all vectors can take on any number and span the space.
What
What is the Image of a Matrix
The set of all possible vectors that can be obtained by multiplying the given matrix by any vector.
How to find the Span of the Basis of the Image of a matrix ?
Solve the matrix down to RREF then identify the column vectors which have a solution given by its pivot in the matrix.
How to find the Span of the Basis of the Image of this particular matrix which has been solved to RREF = 0 ?
All we need to do is identify where the matrix has pivots. in this case it is in the 1st and 3rd columns.
We can then take the original matrix then identify the 1st and 3rd columns as the Vectors that Span the Image of our Matrix
What are the conditions which make Subspace W a Linear Subspace?
A) W contains a 0 Vector
B) W is closed under addition (if V1, V2 is in W, then V1+V2 is in W)
C) W is closed under scalar multiplication
What is the Subspace of a Matrix
A set of vectors that can be formed by taking ALL possible Linear combinations of the Column Vectors
How to show that any Vector in R² is a Linear Combination of Column Vectors
[1 1] & [3 2]
Start by knowing that R² is a Vector we can set the Equation to an addition of the two column vectors time a respective coefficient.
[x y] = α[1 1] +
