Final Exam Flashcards
How can you tell if a T(x) is a linear transformation?
T(0vector) = 0vecter
What is the domain D(T)?
- number of columns of matrix A
- input (pre-image) of the linear transformation
What is the Cod(T)
- R^n space of the image
- R^n space (number of rows) of the matrix A
What is R(T)?
columns space of A
R(T) = Span{v1, v2,…,vn}
the column vectors from the domain (input) of the transformation
How can you determine if a matrix is onto?
1) Put the matrix in REF or RREF
2) Does the matrix have a pivot in every ROW?
If yes, the matrix is onto,
if not, it’s not onto
How can you determine if a matrix is one-to-one?
1) Put the matrix in REF or RREF
2) Does the matrix have a pivot in every COLUMN? If yes, the matrix is one-to-one, if not, it’s not one-to-one
What is an elementary row matrix? (notes 3.2)
If we perform a single elementary row operation on an identity matrix (I), then the result is matrix is called an elementary matrix (E)
What is a property of the elementary row matrix?
If A is any matrix and E is an elementary matrix, then EA is the matrix obtained by performing the corresponding elementary row operation on A.
What is A^T?
A^T is the transpose of A; changing rows to columns of A
What are the operation laws for the transpose matrix?
(A+B)^T = A^T + B^T
(cA)^T = cA^T
(AB)^T = (B^T)(A^T)
IF A = A^T, A is a symmetric matrix
How can you tell if a linear transformation is invertible?
If the linear transformation is both one-to-one and onto, then it is invertible
How can you quickly find the inverse of a 2x2 matrix?
For a 2x2 matrix whose elements are [a b; c d], A^-1 = 1/(ad - bc) * [d -b; -c a]
Mathematically, what is an inverse linear transformation?
T^-1(ȳ) = x̄ if T(x̄) = ȳ
What are the operation laws for invertible matrices?
- A^−1 is invertible, with (A^−1)^−1 = A
- AB is invertible, with (AB)^−1 = (B^−1)(A^−1)
- If AC = AD the C = D.
- If AC = 0nxm then C = 0nxm
What is a property of the inverse matrix?
For a linear system Ax=b, if A is invertible, then we can get a unique solution x = (A^1)y
True or False; Any null-space is a subspace.
True, any null-space is a subspace.
What are two quick ways to identify a subspace?
1) a homogenous (null space) set of equations
2) any span is a subspace
How can you find a basis for a given set of vectors?
1) make the vectors into ROWS of a matrix, A
2) Put the matrix in echelon form
3) the non-zero rows correspond to the original vectors (columns) that form the basis
NOTE: you can do the same thing by leaving the vectors as columns for a matrix A, get the echelon form, but the pivot columns will be linearly independent and the
other columns will be linearly dependent on the pivot columns; the pivot columns correspond to the original columns that form a basis.
What is the rank of a matrix?
After finding the basis of a matrix, you can find the rank.
Rank is the number of vectors in the basis. If there are 2 vectors in the basis, rank(A) = 2
How do you find the basis of a row or column space?
1) Given a matrix (A), put the matrix in echelon form.
2) The non-zero rows correspond to the original rows in A which form the basis of the row space (don’t forget to write these as columns - vectors)
3) pivot columns correspond to the original columns in A that form a basis of the column space
What is the null space and nullity?
Null space are the basis vectors of a homogenous set equations
Nullity is the number of vectors that form the null space
What is the rank-nullity theorem?
rank(A) + nullity(A) = # of columns of A
Given a matrix (A) and a set of vectors, how can you tell if the vectors are eigenvectors of A?
Multiply A by each vector. If the result is c*(the given vector) with the given vector identical to the original vector, then it is an eigenvector and c is the eigenvalue.
When is A diagonizable?
An n × n matrix A is diagonalizable if and only if there are n independent eigenvectors
If A is triangular (upper, lower, diagonal)matrix what does this tell you about it’s eigenvalues?
The eigenvalues of A are diagonal entries.
True or False: An Invertible matrix is not necessarily diagonalizable, and a diagonalizable matrix is not necessarily invertible.
True
